The Reds doubled their lead eight minutes after the restart through Xherdan Shaqiri's cushioned volley and the three points were never in trouble from that moment.
Relive how the action unfolded on Merseyside with Sports Mole's minute-by-minute updates below.
11.00amThis truly is a Super Sunday of Premier League football, with Liverpool, Chelsea and Arsenal all in action before the main event of the Manchester derby. You can follow every kick live with Sports Mole starting at Anfield where Fulham are the visitors, and the team news is in...
11.06amReacting to Liverpool's most disappointing result and performance of the season against Red Star Belgrade, Jurgen Klopp makes four changes as Joe Gomez, Fabinho, Xherdan Shaqiri, and Roberto Firmino are recalled in place of Joel Matip, James Milner, Adam Lallana and Daniel Sturridge.
11.08amThe inclusion of Shaqiri, who was omitted from the trip to Belgrade because of political reasons, gives the Reds an extra attacking dimension in behind the forward three. Sturridge, who has scored four goals in his last three games against Fulham, finds himself back on the bench but there is no place in the squad for Lallana.
11.10amFabinho retains his place to make his third successive Premier League start, having had to wait until October to make his first. Klopp has Jordan Henderson and Naby Keita available again after injury but the Brazilian seems to have earned his manager's trust after a slow start.
11.12amYoung Trent Alexander-Arnold has been going through a slightly iffy period, hooked at half time of Tuesday's Champions League defeat, but Klopp keeps faith in the right-back. Joe Gomez plays centrally with Dejan Lovren only among the substitutes having been suffering with illness this week.
11.14amSlavisa Jokanovic makes three alterations in a bid to change Fulham's fortunes following a sixth successive defeat in all competitions at Huddersfield on Monday. Marquee summer signing Jean-Michael Seri is dropped, with Calum Chambers seemingly deployed in an unfamiliar midfield role.
11.16amLuciano Vietto and Timothy Fosu-Mensah, two more of Fulham's dozen recruits upon promotion, were both substituted midway through the Huddersfield defeat and start from the bench today as Cyrus Christie and Alfie Mawson return to the Premier League' most porous defence.
11.18amHaving begun the season with five goals in six games, Aleksander Mitrovic is now five without scoring although, flanked by Andre Schurrle and Ryan Sessegnon, does still present Fulham with the type of attacking threat that any newly-promoted club from the past decade would be envious of.
11.20amKevin McDonald and Joe Bryan are both missing from the visiting party due to hamstring problems. Injuries and the huge influx of new players has made it difficult for Jokanovic to settle on a nucleus of a starting XI and his inconsistency of selection has no doubt played into Fulham's struggles.
11.22amThe Cottagers have conceded 29 goals in 11 games, the joint-highest figure at this stage of a Premier League season. Southampton let in 29 in the 2012-13 campaign and stayed up, so there is hope for Fulham. Although they needed a certain Mauricio Pochettino to mastermind their escape.
11.24amAs alluded to previously, Jokanovic's constant swapping of defensive shape and personnel must be a contributing factor to that record. The Serbian has yet to name the same back four in successive league games this season, with a total of nine players having been used across the back line.
11.26amFurthermore, Jokanovic has already used three different goalkeepers, all of whom have made at least two Premier League starts. Marcus Bettinelli has lost his place as number one, but not in the England squad, so Sergio Rico, on loan from Sevilla, is the man tasked with keeping out Liverpool today.
11.28amRico was one of 12 players brought to Craven Cottage in an ambitious summer recruitment drive. Of those dozen, only Andre Schurrle has arguably had the desired impact, with four goals in 10 Premier League starts showing that the German World Cup winner remains a classy finisher.
11.30amAlthough the goals have dried up in recent weeks, it has definitely been at the other end of the pitch where Fulham have had their biggest problems. The Cottagers, alongside Macclesfield Town who are bottom of the entire Football League, are the only English clubs yet to keep a clean sheet this season.
11.32amDefeat today and Fulham could equal their club record of six consecutive Premier League defeats, set in 2001-02 and 2012-13. Both of those streaks included a loss against Liverpool, albeit at Craven Cottage, not at Anfield.
11.34amThat is not to say that Fulham have enjoyed much success on Merseyside either. The club's only win in 54 away league games against Liverpool or Everton was a 1-0 victory at Anfield in May 2012, Clint Dempsey scoring an 85th-minute goal after Jay Spearing was sent off for the Reds.
11.36amIn fact, Fulham's record against the Premier League's established top six is generally poor, taking just one point from their last 18 matches against England's elite. That includes three defeats this season - to Man City, Spurs and Arsenal - in which they have conceded 11 goals and scored twice.
11.38amAt the other end of the spectrum, Liverpool have eaten newly-promoted teams for breakfast under Jurgen Klopp, at Anfield at least, winning all eight of their Premier League home games against teams to come up from the Championship, scoring 25 goals and conceding just three.
11.40amMohamed Salah has been key to Liverpool's dream killing, averaged a Premier League goal or assist every 66 minutes against newly-promoted teams since joining Liverpool including a goal and two assists for Sadio Mane when Cardiff City visited Merseyside a fortnight ago.
11.42amHowever, it is not just against newly-promoted sides that Liverpool have been dominated at home in recent times. The Reds are in the midst of a 26-match unbeaten top-flight run at Anfield, their longest such run since a 31-match streak under Rafael Benitez from 2007 to 2009.
11.44amVictory today would send Liverpool to the top of the Premier League table ahead of this afternoon's Manchester derby, and yet Jurgen Klopp has admitted that his team have not been at their explosive best.
He is quoted by BBC Sport as saying: "The season so far-wise, points-wise in the Premier League, has been really successful. But it doesn't really feel like that because there are two reasons in comparison with last year: free-flowing football in a few games and Man City and maybe Chelsea and a few others play a very good season as well."
11.46amWhile the Reds have not been dispatching of teams with the same aplomb as Man City, they have taken great strides to catch up with their title rivals defensively. If they keep a clean sheet today they will equal the club record for fewest goals conceded - five - after 12 top-flight fixtures.
11.48amIn a sign that Liverpool have dipped below their best, last weekend's 1-1 draw with Arsenal saw them finish with their lowest possession stats recorded in the Premier League since February's 2-2 draw with Tottenham, at 38.2%.
11.50amThe Reds have actually only won three of their last nine games in all competitions, and the midweek setback against Red Star has left their European campaign in peril. Klopp has made it clear where Liverpool's priorities lie this season, though, telling BBC Sport: "It feels a bit like this season can only be a success if we become champions."
11.52amHEAD TO HEAD: Liverpool have won the last four meetings between these sides, scoring a total of 14 goals. In contrast, the Londoners have scored just four times in their last 17 trips to Anfield.
11.54amWhile the pressure is rising on Jokanovic - leading bookmakers make him the 8/11 favourite to be the first Premier League manager sacked - today is a bit of a free swing for the Cottagers. However, if they come out and try to outplay Liverpool, they could find themselves on the end of a drubbing.
11.56amThe players have emerged onto the field early in preparation for the Rememberance Day tributes, 100 years on from the end of the First World War. You'll Never Walk Alone is ringing round Anfield at present, but soon the ground will fall silent to honour all of our fallen troops.
11.58amPREDICTION: This really looks like the home banker of the Premier League weekend. I can't see anything other than a Liverpool win, the question is; by how many? Fulham could be on the end of a backlash from the Red Star game so I'm going to say the Reds put at least four past them.
1 minKICKOFF: The action is underway, with Paul Tierney the man in the middle this afternoon...
3 minMitrovic draws first blood on Van Dijk with a cheeky nutmeg as Fulham work the ball nicely down the left. Fabinho is on hand to intercept just as Tom Cairney was getting into a shooting position.
4 minCHANCE! Gomez switches the play to Alexander-Arnold, who whips in an early cross which is just behind Firmino, making it difficult for the Brazilian to bring under control. The ball drops perfectly to Mane at the back post but the Senegalese scuffs his shot horribly wide.
6 minLiverpool again attack down the right, with Alexander-Arnold knocking a cross behind for a goal kick. It was a poor final ball but the full-back should have been teed up earlier by Shaqiri who held onto it for too long.
8 minThe inclusion of Chambers as a holding midfielder is an intriguing selection from Jokanovic. It is not a position that the youngster has played in, certainly with any frequency, at his previous clubs, but it does give the Cottagers some flexibility of shape, and more defensive awareness.
10 minMitrovic earns Fulham their first corner of the game but will be frustrated that Schurrle couldn't beat the first man with his delivery.
11 minMane skips past a Fulham defender and tries to release Salah down the middle but just overhits his through-ball.
13 minCLOSE! Space opens up nicely in the inside right channel for Shaqiri who shoots narrowly wide from 25 yards out, and in fact Rico did get a touch on it so it should have been a corner. The switch to the right flank is on for Liverpool all the time with Sessegnon failing to track back.
15 minSAVE! Firmino and Salah work it wonderfully through a tight area, and Odoi slides straight past the Egyptian to leave him one-on-one with Rico, who stands up tall and blocks at point-blank range.
17 minThe hosts have pushed Fulham back in the last five minutes and that shaky defence is starting to be tested. Mitrovic has a hugely important role in holding the ball up to give them respite whenever possible.
19 minCHANCE! Again, Sessegnon does not track Alexander-Arnold who runs in behind the teenage prodigy, to meet Shaqiri's cross. He attempts an ambitious header from the right edge of the box and it drops just over the bar.
21 minSAVE! Shaqiri drops a lovely ball over the top for Salah who hits it sweetly on his right foot, but Rico gets a strong hand to keep out the half-volley. The goalkeeper would have been disappointed to concede from there but it was a decent save nonetheless.
23 minCHANCE! What a chance for Fulham, with Mitrovic winning a flick-on and Sessegnon getting a slice of luck to get through Gomez. The youngster is through on goal, takes his shot early but pulls it a couple of yards wide. The Cottagers won't have many chances so that is a big moment in this game, you feel.
25 minShaqiri is taking up a lot of dangerous positions in between the lines. Chambers needs to keep an eye on the Swiss but Fulham seem determined to hold their shape and not follow the ball.
27 minSHOT! Mane is given too much room to get a shot away on the edge of the box but can't keep it down.
29 minSAVE! Another decent opportunity for Fulham as Schurrle lets fly from 25 yards out. The shot bounces in front of Alisson who parries it straight back into the danger zone, and is lucky that Gomez is there to bail him out before Sessegnon or Mitrovic can pounce.
31 minFeint shouts of a penalty from Liverpool as Le Marchand barges into the back of Salah, but the ball was out of the Egyptian's control and the Fulham defender simply couldn't slow down in time.
33 minFulham's defence is holding up well so far, blocking Shaqiri's free-kick at the top of the ball and then rushing out to clear the second ball through Cairney, whose return is most welcome.
35 minSHOT! The hosts are not having this as easy as most would have expected. Sessegnon skilfully turns away from Van Dijk nicely but then loses balance and blazes high into the stands.
37 minSHOT! Much of Liverpool's threat has been down their right, but this time Mane gets to the left byline, and his pull-back goes through a crowd of bodies to reach Shaqiri on the edge of the box. The former Stoke winger tries to generate too much whip on his left-footed shot and drags it wide.
39 minAgain, Mane gets away down the left channel but a heavy touch allows Mawson to come across and clear. That was the first time that Fulham had shot themselves in the foot defensively, a mix-up between Christie and Schurrle, and the Senegalese winger really should have done better.
41 minAn unbelievable 15 seconds at Anfield, with Mitrovic seeing a goal disallowed for Fulham and Liverpool racing up the other end to score. The Cottagers are fast asleep after the flag goes up, and two passes later Salah is through on goal to slip it through the legs of Rico.
43 minSAVE! Fulham continue to have their fair share of chances, with Cairney teeing up Mitrovic for a shot which Alisson gets down to, low to his right. The Serbian couldn't generate much power on his weaker right foot but that was a good opening 12 yards from goal.
45 minSessegnon tries his luck from 30 yards out but it is easy pickings for Alisson who quickly starts another Liverpool attack. The goalkeeper deserves credit for his quick thinking for the goal, playing it out to Alexander-Arnold who released Salah down the right.
12.50pmLiverpool head into the break with a 1-0 lead over Fulham, but it arguably should be the other way around. Mitrovic was flagged offside as he headed home Cairney's cross, although replays showed the striker to be level with Andrew Robertson. Salah went straight up the other end to score. When it isn't going your way, it isn't going your way...
12.52pmThe rest of the first half didn't play out as may have been expected, with Fulham's leaky defence standing up well to Liverpool's attacking play. The Reds still created a fair few chances, but so did Fulham who could have gone ahead through Sessegnon earlier in proceedings.
12.54pmWhat will increase Fulham's frustration over the Salah goal is that the ball was still rolling when Alisson started the counter-attack. Yes, it is a little pedantic, but the ball had not been spotted properly for the free-kick for the offside decision, and that is the letter of the law.
12.56pmWhile the away dressing room may be a bit downbeat because of the circumstances of the goal, Fulham should be encouraged by their first-half display. Their defensive shape looked a lot more solid than it has in recent weeks and they posed a threat through the front three.
12.58pmA warning for Liverpool, and a glimmer of hope for Fulham to hang onto...
46 minKICKOFF: Liverpool get us underway in the second half, with no changes from either side at the break...
48 minWe did not see much of Robertson in the first half, with Liverpool's attack being focussed down the right. The Scotsman is reportedly close to agreeing a new contract to double his wages at Anfield.
50 minA theatric fall from Shaqiri buys Liverpool a free kick to the right of the penalty area. You'd have fancied the Swiss to take it, but Salah steps up and knocks what was neither a cross nor shot straight out of play.
51 minGomez is lucky to get away without a booking for cynically wrestling back Mitrovic as Fulham threatened to break away. Referee Paul Tierney is yet to go to his pocket today but should have there.
52 minSAVE! Mane connects beautifully with a 25-yard strike which Rico tips over the crossbar at full stretch.
53 minFulham may be cursing their luck but they have been let down by another lapse of concentration at the back. Shaqiri is left unmarked at the back post and takes advantage with a lovely cushioned volley to send Robertson's looping cross back across Rico and into the bottom corner.
55 minIt is a long way back for the Cottagers now and you fancy Liverpool might move through the gears soon...
57 minLe Marchand tries to release Sessegnon down the left channel but Alisson, now sporting a the rarely-spotted goalkeeping cap, is off his line quickly to sweep up the danger.
59 minYELLOW CARD! Referee Tierney now does brandish his first card of the game, to Chambers who caught Salah on the ankle. The Arsenal loanee wouldn't have had many complaints if that was red after sliding in studs up.
62 minSAVE! Firmino picks up the ball in plenty of space on the halfway line and travels with the ball. Mane makes a good decoy run to allow the Brazilian to release Robertson on the overlap, and the Scot gets his effort on target but Rico makes a good save to his left.
64 minSUBSTITUTION: Cairney has had a decent game for Fulham and will be a key player for them moving forward. Jokanovic is still managing his fitness though so Seri is on for the remainder.
66 minIt looks a lot more comfortable for now for Liverpool, who are seeing a lot of the ball in the Fulham third and are no longer being posed with the same sort of threat that the visitors showed before the break.
68 minSalah's quick feet earns Liverpool a free kick, central and 20 yards from goal. Alexander-Arnold sends the set piece into Seri's head, and the Fulham player is dazed and requires some attention.
70 minSUBSTITUTION: Klopp takes this opportunity, leading 2-0 at home, to get some minutes under Henderson's belt as the skipper replaces Wijnaldum, and is immediately handed the armband by Van Dijk.
72 minFulham are being dragged around now and are being pulled out of shape by Liverpool's quick passing. It is the Cottagers who look tired, not their hosts despite their midweek European excursions.
74 minRobertson may have been second fiddle to Alexander-Arnold in the first half, but the Scotland international has arguably been Liverpool's most prominent attacking threat since the restart, and already has an assist to his name.
76 minYELLOW CARD! Gomez should have been booked earlier in the half and is now, for tugging on the shirt of Mitrovic who had shown good chest control to initially get the wrong side of the England defender.
78 minSUBSTITUTION: Jokanovic makes his second change as Luciano Vietto replaces Schurrle, who has been quiet in the second half.
80 minFirmino gets in a dangerous position by the left byline, but takes a little too much time on the ball and allows Fulham to get back into shape. The eventual comeback to Alexander-Arnold leads only to a wild shot over the crossbar.
82 minSUBSTITUTION: Not that there has been much threat from Fulham, but Klopp is going to tighten things up for the final 10 minutes with James Milner coming on for the excellent Shaqiri, a man of the match candidate.
84 minSUBSTITUTION: And there is also a change in midfield for Fulham as Anguissa is swapped for Stefan Johansen.
86 minSHOT! Christie makes a good run down the right and pulls the ball bag for Mitrovic, who lets the ball roll across him but then slices his shot wide.
88 minThis game has been dead and buried since the second goal and Liverpool are going to the top of the Premier League table. Fulham meanwhile, will stay rooted to the bottom after six straight defeats.
90 minSalah tries to bend one inside the far post but Mawson makes a block on the penalty spot.
90+2 minSUBSTITUTION: Naby Keita is welcomed back from injury by the Anfield crowd as Fabinho trots off, job done.
1.55pmIt was billed as a home banker, and that's what it turned out to be in the second half as goals from Salah and Shaqiri earn Liverpool all three points and put the pressure on Chelsea and Man City this afternoon. Fulham will argue it could have been different if not for a controversial disallowed goal...