Marco Silva's impressive visitors looked to have battled their way to a goalless draw before Jordan Pickford's stoppage-time misjudgement handed the Reds all three points.
Pickford pushed a skied volley from Virgil van Dijk that appeared to be dropping over his crossbar back into play, and Origi pounced with a header from a couple of yards out.
Yerry Mina and Andre Gomes missed first-half chances for Everton who should be encouraged by their display but remain winless in the Premier League at Anfield since 1999.
Relive how the action unfolded with Sports Mole's minute-by-minute updates below.
3.15pmFor the 100th time in league action at Anfield, Merseyside rivals Liverpool and Everton do battle this afternoon in one of English football's most historic and fiercest derbies. Who will claim bragging rights and three precious points? Stay tuned with Sports Mole's minute-by-minute updates to find out.
3.21pmJurgen Klopp makes three changes from the Champions League defeat to Paris Saint-Germain, with Jordan Henderson missing through suspension and Dejan Lovren and James Milner also dropped. Local lad - these fixtures aren't the same without them - Trent Alexander-Arnold is one of those recalled.
3.23pmSummer signings Fabinho and Xherdan Shaqiri are also given the biggest show of faith in them yet by Klopp. The two players had only made seven Premier League starts between them this season prior to today, but are thrown in at the deep end in the Merseyside derby. Will they sink or will they swim?
3.25pmShaqiri's inclusion gives Liverpool some tactical flexibility. The Swiss international has excelled in his Reds appearances so far as a central midfielder with freedom but he is unlikely to get that in the cauldron of a derby, so it could be a more conventional 4-2-3-1 with Shaqiri in behind Roberto Firmino.
3.27pmEverton boss Marco Silva names an unchanged starting XI from the one which edged past Cardiff City at Goodison Park a week ago. His only selection dilemma in recent weeks has been Kurt Zouma or Yerry Mina at centre-back, and the giant Colombian has retained his place over the Chelsea loanee.
3.29pmRicharlison continues as Everton's number nine with Cenk Tosun rooted to the bench these days. The Brazilian has started to justify his £40m price tag with six goals already this season, a return that is matched by Gylfi Sigurdsson in a Toffees attack which has a variety of weapons to hurt the opposition.
3.31pmIn a mark of how much Silva has put his own stamp on this Everton team, there are only three survivors from the Toffees team which started this fixture last December - Jordan Pickford, Idrissa Gueye and Sigurdsson. Mind you, there are only four Liverpool survivors (Gomez, Salah, Mane and Robertson).
3.33pmA key battle this afternoon could be that of former Roma teammates Salah and Lucas Digne, who has usurped the long-standing Leighton Baines as Everton's first-choice left-back. Digne told ESPN of the Salah challenge this week: "I play football to play against the biggest players in the world."
3.35pmThis could be the perfect fixture for the hosts to take out their frustration from the midweek defeat to the Amateur Dramatic Society of Paris. It was a result that left their Champions League campaign in jeopardy, but the Reds continue to look imperious domestically...
3.37pmLiverpool are unbeaten in this season's Premier League, and have 33 points, which is their highest ever tally at this stage of a Premier League season. The only games in which they have dropped points have been against Manchester City, Chelsea and Arsenal, but the former's relentess pace at the top of the table still sees them five adrift.
3.39pmAnfield has not only been a graveyard for Everton recently, but the last 27 visitors in the Premier League. The Reds have won have won five of their six home games at Anfield in the league this season, and only conceded one goal, against Cardiff City.
3.41pmThis will be Mohamed Salah's 50th league appearance for Liverpool, in which he has scored 39 goals. One more strike today and he will match the club record of 40 set by George Allan from 1985-1987. We know that Salah takes huge pride in this records, telling Sky Sports this week: "it was in my head 24/7," speaking of his all-time Premier League-best 32-goal haul from last term.
3.43pmIn the history of the Premier League, only two players have netted more in their first 50 Premier League games for a single club: Alan Shearer and Andy Cole, who scored 41 for Blackburn and Newcastle respectively. What odds on a Salah hat-trick today?
3.45pmPerhaps the only player of the current squad more adored than Salah is Virgil van Dijk. The Liverpool fans have had a love affair with the Dutchman ever since his 84th-minute FA Cup winner against Everton on his Reds debut. Van Dijk has a reputation as a goalscoring defender, and has bagged twice for the Netherlands of late, but has not found the net in 39 club appearances since.
3.47pmOf their century of Anfield encounters, Liverpool boast 43 wins, with 33 draws and 23 Everton victories, although none for the Toffees since way back in 1999. Will Liverpool's dominance continue today or can Marco Silva change the tide?
3.49pmSilva is aiming to become the first Toffees boss since Joe Royle, who was unbeaten in seven Merseyside derbies, in November 1994 to win his first Merseyside derby in the Premier League. No Everton manager has ever won their first league match at Anfield.
3.51pmHowever, Silva does have a good track record against Liverpool at his previous clubs, leading Watford to a 3-3 draw on the opening day of the 2017-18 season, and Hull City to a 2-0 win over the Reds in which current Everton striker Oumar Niasse was on the scoresheet, although both of those results were at home.
3.53pmWhile Everton's recent record at Anfield puts most of the pressure on title-chasing Liverpool, Klopp has tried to redirect some of that on his opposite number by telling BBC Sport that Silva "has the best squad" of any Toffees manager he has faced. "It will be a proper fight but we expect that in the derby. That's how derbies should be," he added.
3.55pmAnd it may be true that Everton visit their neighbours this afternoon with a little bit more hope than usual. The Toffees have won five of their last seven top-flight games with their solitary defeat in that run coming at another unhappy hunting ground in Old Trafford.
3.57pmSilva, first courted by Everton after Ronald Koeman's sacking in October 2017, is starting to prove to Toffees fans that he is worth the wait. The Blues won only one of their first six league matches of the season but are now 6th in the table and one of the most improved teams in the league.
3.59pmThe likes of Richarlison and Gylfi Sigurdsson may take most of the headlines, but it has arguably been in defensive areas where Everton have made the biggest strides. After shipping 11 times in their first six league games under Silva, they have only conceded four in the subsequent seven outings.
4.01pmOne of the next targets for the former Sporting Lisbon manager will be to improve his side's away form. Last season, only the three relegated clubs recorded fewer than Everton's three Premier League wins on the road, and this term their only success has been a 2-1 victory at Leicester City in October.
4.03pmRESULT! A thrilling North London derby has just come to an end with Arsenal coming from 2-1 down to run out 4-2 victors. Pierre-Emerick Aubameyang was the star of the show with two goals and an assist for the Gunners who go above Spurs in the table with this triumph.
A reminder that earlier today Chelsea moved to within two points of Liverpool with a 2-0 win over Fulham.
4.05pmAround 10 minutes until kickoff at Anfield. I'm sure it will get feisty after the whistle but Jurgen Klopp - and Seamus Coleman - have taken measures to unite the blue and red halves of Merseyside prior to kickoff.
4.07pmHEAD TO HEAD: Liverpool are unbeaten in the last 15 league meetings with Everton home and away, although nine of those have been draws. Their last defeat came in October 2010 during Roy Hodgson's ill-fated spell in charge.
4.09pmEverton's best chance this afternoon might be to make this an old-fashioned sort of derby, On their last win at Anfield, there were three red cards with Sander Westerveld, Francis Jeffers and a 19-year-old Steven Gerrard were all sent off after Kevin Campbell's early goal.
4.11pmThe gap to Man City stands at five points prior to kickoff and Liverpool can't afford to lose ground at home. Everton will stay sixth regardless of the result today but a win would mean so much more than three points - it would show the Reds aren't the only team on Merseyside making waves.
4.13pmPREDICTION: Recent Everton teams have been beaten before they've arrived at Anfield but I don't think this Silva outfit will be shy of confidence. If they can survive the first 20 minutes without too much damage they could push the Reds all the way, but I do think the hosts will prevail 2-1.
1 minKICKOFF: The action is underway at Anfield, where Chris Kavanagh becomes the least experienced referee to take charge of this combustible fixture. It is a nervy start from Jordan Pickford who boots a clearance straight out of play to sarcastic cheers from the home crowd who are unsurprisingly rocking.
3 minCLOSE! Everton are mere yards away from taking a shock lead, with Digne delivering a free kick to the back post where Mina had wriggled away from Van Dijk to head back across goal, but just wide. The ball bounced into the floor and Alisson looked to his right in desperation as it drifted past the post.
5 minThe shape that Liverpool have started with is very interesting, with Firmino playing in a very withdrawn role and Salah furthest forward in the inside right channel and Shaqiri trying to drift into pockets in the middle. The movement of that front four could pull Everton this way and that if they aren't careful.
7 minLiverpool work it right to Alexander-Arnold and Digne is needed to slide in and block his cross behind for a corner.
9 minThe Toffees came here last year under Sam Allardyce and sat 10 men behind the ball, but there is no sign of that approach from Silva whose boys are trying to play through midfield and get the ball into the feet of their skilful attack.
11 minCHANCE! Now Liverpool have a golden chance as Salah turns the ball around the corner for Mane who blazes over from 12 yards on the volley, a difficult effort to control on the bounce. The Reds have gone through the gears in the last couple of minutes with a couple of crosses just out of reach.
13 minCHANCE! Gomez knocks an early ball over the top for Mane, who had got the wrong side of Coleman. The Senegalese is forced onto his weaker left foot and his shot from 12 yards is blocked by a combination of Coleman and Pickford.
15 minMane has started this match in lightning form, pressing relentlessly to help Liverpool win the ball back in the Everton half and showing some silky touches with the ball too. The Toffees are playing a dangerous game by playing out from defence against this Liverpool press.
17 minGomes slips a pass inside for Richarlison who skips by Van Dijk and goes to ground. The Brazilian may have been anticipating contact but Van Dijk pulled his leg back in so there was definitely no penalty, and to be fair the Everton man showed no interest in one. If he'd have managed to stay on his feet he'd have had a great chance to get a shot away inside the box.
19 minSHOT! Salah feeds the ball into the feet of Firmino who turns in the box but fails to generate any power on his left-footed effort.
21 minSAVE! That is what you pay the big bucks for! It is a brilliant save from Alisson to spread himself and block a Gomes header from point-blank range. Everton had worked it nicely with Walcott heading Bernard's cross back into the centre, and it looked as though Gomes couldn't miss from six yards out. Gomez deserves credit to for reacting quickly to clear the loose ball as it was threatening to sneak over the line.
23 minSet pieces are looking promising for Everton, who are going to lengths to try to block off Van Dijk and allow Mina a free run into the danger zone. The delivery from Sigurdsson this time is very heavy so the visitors don't get a chance to see their plan come to fruition.
25 minDelightful footwork from Gomes who drags Fabinho all over the place on the left flank doesn't lead to much, but it does show the confidence with which Everton are playing. There are some proper footballers out there in blue shirts - Gomes, Bernard, Digne all very comfortable on the ball.
27 minOne of those fine ball-players - Bernard - is perhaps guilty of overplaying as he gets pushed back to his own corner flag by Alexander-Arnold's pressing, and takes the ball behind for a corner from which Michael Keane takes it off the head of Van Dijk with an important clearance.
29 minSHOT! Firmino is allowed to drift into a pocket in between the midfield and defence and gets the ball into feet from Robertson. The Brazilian turns and tries to bend one into the top corner from 30 yards out but it is easy pickings for Pickford.
31 minEverton have the chance to catch Liverpool short at the back, and Walcott does well to knock the ball onto Richarlison in the left channel. The former Watford forward knows he has Alexander-Arnold down his neck so shapes to take his shot on early, but TAA beat him to the ball anyhow as he threatened to pull the trigger.
33 minYELLOW CARD! Shaqiri is the unlikely recepient of the game's first yellow card for a cynical trip on Bernard who was running clear down the Everton left.
34 minSAVE! Fabinho pinches the ball off Gueye on the halfway line and Salah quickly releases Shaqiri with a through-ball out of Mina's reach. The Swiss is through one-on-one but never looks comfortable on his right foot and Pickford blocks his shot wide of the post. Another golden opportunity - it could be 2-2!
36 minWhat a pass from Sigurdsson, who picks out the run of Walcott with a reverse through-ball. The ex-England international tries to round Alisson and his first touch isn't a bad one, but the Liverpool goalkeeper stretches to knock the ball off Walcott and behind safely for a goal kick. This is end-to-end stuff.
38 minMane tries to find Alexander-Arnold with a raking cross-field pass but it is just out of the full-back's reach. Alexander-Arnold has been a constant outlet for Liverpool but we have not seen much of Robertson as an attacking force so far.
40 minPickford is very quick off his line to boot clear a ball over the top which Mane was racing on to.
42 minThe passing of Gueye has been a bit of a liability for Everton at times in the first half. He is not as composed as Gomes on the ball and if Liverpool continue to press him they might find some joy.
44 minSalah nearly squeezes through a couple of Everton bodies by the left byline but Coleman stands up strong. The Egyptian King has stretched Everton with his running but hasn't had a sight of goal yet.
45+1 minGomes shows good skill again - and strength - to create half a yard against Wijnaldum and get a low cross in. Sigurdsson could have made a run to the near post but as it is, it is a simple claim for Alisson.
5.04pmIt has been a long time since I've seen an Everton side come to Anfield and give as good as they get, but that is exactly how it was in the first half. It has been an even Merseyside derby thus far, with both sides squandering a couple of glorious chances each. Finely poised for the second period!
5.06pmLet's remind ourselves of the best opportunities from that half, starting with Mina who was inches from putting Everton ahead with third-minute header. Gomes also had a headed chance from four yards out superbly blocked by Alisson. At the other end, Pickford saved one-on-one with Shaqiri.
5.08pmRESULTS: A reminder that today's earlier Premier League games saw Chelsea deal Claudio Ranieri his first defeat as Fulham manager, and Arsenal beat Spurs 4-2 in a blockbuster North London derby. You can read our match reports on those encounters in our fixtures & results section.
5.10pmBENCH WATCH: Marco Silva will be very pleased with what he has seen, although Idrissa Gueye has struggled with the pace of the midfield passing and pressing so I wonder if Tom Davies is an option he is considering. Firmino has looked uncomfortable in his deeper role for Liverpool, so it may be time for Naby Keita to show us what he is capable of?
5.12pmHistory suggests this one will finish as a draw but I'd be amazed if there wasn't at least a couple of goals in this game.
5.14pmSTAT:Mohamed Salah had fewer touches in that first half than his goalkeeper Alisson. The prolific winger was a little on the periphary but you wouldn't bet against him changing that and becoming the match-winner.
46 minKICKOFF: The action resumes at Anfield with both sides unchanged for now; Liverpool attacking the Kop end...
48 minSHOT! Salah is given the time and space to turn outside the area and accepts the invitation to shoot, using Firmino as a shield to blind Pickford with a curling shot which finishes just wide of the post.
50 minLiverpool nearly work a wonderful chance for Firmino, and it takes a scrambling Bernard clearance to take the ball off the Brazilian's foot after Alexander-Arnold's cushioned layoff, which was a little behind his teammate.
53 minCHANCE! How is this game still goalless? Mane should score, played through on goal by Firmino after Keane had abandoned his defensive position to try to steal the ball. Pickford makes himself big at the near post but Mane can't even hit the target 12 yards out to the left of goal.
55 minThat looks like a nasty one for Walcott, who is caught on the Achilles by Wijnaldum. Ademola Lookman has had some bright cameos off the bench but Silva may not have wanted to use him this early.
57 minWalcott looks as though he will be okay to continue, but another Everton forward is in the wars now with Richarlison taking a big bump from Fabinho, who sliced his international teammate down.
59 minSHOT! Mane rolls the ball into Salah's feet and the Egyptian skilfully returns it to the Senegalese, whose bending effort from 25 yards is wide of Pickford's post.
61 minSalah doesn't quite hold his run and is flagged offside from Mane's pass, with Pickford making the save regardless.
63 minTempers are starting to flare a little with this derby in the balance. Robertson antagonises Richarlison with a needless scuffle over a throw-in, and referee Chris Kavanagh deals with it very sensibly.
65 minSUBSTITUTION: Silva has blinked first, although his hand may have been forced by that knock to Walcott who mays way for Lookman.
67 minSHOT! Lookman drives forward, taking a couple of flagging Liverpool players out of the game, and shifts the ball on to Bernard. The Brazilian cuts inside but doesn't catch his 20-yard shot quite right and it swerves right of Alisson's left-hand post.
69 minRobertson's cushioned cross on the volley is into a dangerous area but just out of the reach of Mane.
71 minSUBSTITUTION: Klopp makes his first change, bringing on Naby Keita in place of Shaqiri, who may be disappointed to have been hooked with Firmino arguably the least effective of the front four today.
73 minLookman has settled into the game quickly and has already showed some confident touches. Klopp will be hoping for the same impact from Keita whose start at Anfield, albeit injury disrupted, has been underwhelming.
75 minSUBSTITUTION: Big call from Klopp, who brings of Salah for the final 15 minutes and replaces him with Daniel Sturridge. There are lots of games coming up in December but surely you keep Salah on if you're trying to win a derby?
77 minYELLOW CARD! Firmino gives the ball away on the halfway line and Fabinho ensures Everton don't take advantage with a tactical foul.
79 minYELLOW CARD! Gomez becomes the third Liverpool player to enter the book for clattering into the back of Richarlison who has done a great job in occupying Gomez and Van Dijk all afternoon.
81 minCHANCE! Liverpool counter from an Everton set piece, but the opportunity falls to the wrong man as Firmino does not have the pace to get away from Coleman, and eventually drags a left-footed shot wide.
83 minSUBSTITUTION: Final throw of the dice for Klopp as Divock Origi comes on for his first Premier League appearance for Liverpool in nearly 18 months. Firmino makes way after a poor game.
85 minAlexander-Arnold's corner is cleared at the near post by Digne. Set-piece delivery from both teams today has been average.
87 minGood play from Origi who rolls away from Mina and uses his fresh legs to run clear of the Colombian. His cross is into the perfect area as well but Coleman does brilliantly to beat Mane to the ball at the expense of a corner.
88 minOFF THE BAR! Origi could have been the unlikely hero for Liverpool! Alexander-Arnold gets a corner spot on and Van Dijk wins the initial header which drops for Origi five yards from goal. Although he is so close to goal it is a tricky one for the Belgian who flicks the bouncing ball against the top of the crossbar!
90 minSUBSTITUTIONS: Silva is settling for a draw now and has wound the clock down with successive substitutions. Dominic Calvert-Lewin and Kurt Zouma are on for Sigurdsson and Bernard. Four minutes of stoppage time.
90+2 minAnfield calls for a penalty on Sturridge but it is a fair tackle from Richarlison, in his own area in stoppage time. The substitute had skilfully evaded Gomes but waited too long to get his shot away.
90+4 minEverton's physio is on to give some treatment to Gueye which ticks away a few more precious seconds...
90+6 minGOAL! Divock Origi has just written himself into Liverpool folklore with the latest, the craziest of Merseyside derby winners! Everybody else in the ground had given the ball up after Van Dijk had skied his volley high in the air, but the Belgian put the pressure on Pickford who makes the unfathomable decision to play the ball, which was dropping over his own crossbar. Pickford succeeds in only pushing it right onto the head of Origi who can't miss from two yards out.
6.13pmWhat an extraordinary finish to this Merseyside derby! Jurgen Klopp stormed the pitch upon Origi's goal to celebrate with his players and his emotion was shared by the majority inside Anfield. Everton had held their own for 96 minutes but a huge error from Pickford has thrown a point down the drain.
Liverpool won't care - it is an 11th win from 14 for them and they stay in touch with Manchester City.
6.15pmThat's it from our bumper Sunday coverage of the Premier League. Relive the dramatic closing moments of this Liverpool victory with our match report here, and catch up on the earlier action here, with Chelsea and Arsenal the other big winners today. Thanks for joining us. Until next time, goodbye!