West Ham defended firmly for much of the game and only afforded the Blues with one clear-cut equalising chance, which Morata fired into the side-netting late on.
Relive how the action unfolded courtesy of Sports Mole's minute-by-minute updates below.
11.30amHappy Saturday ladies and gents! It is a massive weekend of derbies in the Premier League and Sports Mole has you covered all the way through, starting with this all-London clash between West Ham United and Chelsea at the London Stadium. Team news coming up...
11.36amAs widely expected, Joe Hart is dropped as Adrian keeps his place in goal after a valiant performance in defeat to the England number one's parent club Manchester City last weekend. That is only going to increase the scrutiny on Hart's position with the Three Lions ahead of the World Cup.
11.38amDavid Moyes has made three changes from that 2-1 defeat at the Etihad Stadium, one of which sees Winston Reid return from a hamstring injury to replace youngster Declan Rice at centre-back. Jose Fonte is still out though, meaning that last week's goalscorer Angelo Ogbonna is Reid's partner.
11.40amElsewhere, Mark Noble is back for a derby fixture which he will love, while big-money signing Marko Arnautovic provides some extra flair in an attacking midfield three. Cheikhou Kouyate is not involved today because of a hamstring strain, while Edmilson Fernandes is demoted to the substitutes.
11.42amMoyes's game-plan will be pretty clear - to sit back and frustrate Chelsea and try to pinch something from the game late on. His bench is well suited to that approach, with the fit-again Javier Hernandez, Andy Carroll and Andre Ayew all in reserve to make an impact in the second half.
11.44amThe versatile Michail Antonio will lead the Hammers line, and he certainly has the pace and power to cause Chelsea's defence problems. His lack of goals be a slight concern - the former Nottingham Forest man only has one this season compared to six at this stage last campaign.
11.46amConte resists the temptation to make wholesale changes after Tuesday's Champions League draw with Atletico Madrid, which secured the Blues' passage to the last 16. Marcos Alonso for Victor Moses is the only alteration, with Eden Hazard continuing in the No.10 role behind Alvaro Morata.
11.48amConsidering the brilliant attacking outlet that he offered them on his Premier League return against Newcastle last weekend, Moses's removal from the starting lineup is likely to be fitness related following his recent two-month layoff with a hamstring problem. Alonso offers just as much going forward down the left, in fairness, with Davide Zappacosta switching back to the right flank.
11.50amDanny Drinkwater misses out on the Chelsea squad today because of illness, which means there is a space for 17-year-old midfielder Ethan Ampadu on the bench. He is yet to appear for the Blues in the Premier League but has played twice in the EFL Cup this season.
11.52amChelsea have won six of their last seven top-flight games and key to this form has been the performances of Hazard and Cesc Fabregas. Conte has identified these two as his most talented players and adjusted his system to give them the best chance of influencing matches.
11.54amHazard has certainly responded to his new position by rediscovering his title-winning form. The Belgian has five goals in his last six games - a timely warning to England following the World Cup draw.
11.56amThe 26-year-old has an excellent record against the Hammers too. Hazard has been involved in seven goals in his last eight Premier League games against West Ham, scoring four and setting up three.
11.58amAlvaro Morata has been sharing the attacking burden with Hazard and has been equally important for Chelsea this season. The Spaniard has been involved in 13 goals for Chelsea in his first 14 Premier League appearances, scoring nine times and supplying four assists.
12.00pmMorata's next goal will be his 50th in top-flight football across spells at European giants Real Madrid, Juventus and Chelsea. As an unbiased Fantasy Football manager, I wouldn't mind if that came at the London Stadium today!
12.02pmSTATS: The Blues have scored a league-high nine headed goals this season. Morata has five of those, largely thanks to the service of compatriot Cesar Azpilicueta. No combination has been more productive this term.
12.04pmChelsea traditionally perform very well in December, winning each of their last seven matches in the Christmas month. They need that festive form to continue if they are to keep in touch with leaders Manchester City, who are 11 points ahead at the start of this weekend.
12.06pmThe defending champions have a mixed record in London derbies this season, beating Tottenham and drawing with Arsenal, but losing to rock-bottom Crystal Palace in their last league defeat in October.
12.08pmFour games into Moyes's tenure, the Scotsman has yet to make any significant improvements to West Ham's fortunes. The Irons have picked up only one from a possible 12 and remain in the relegation zone. Even victory today would not lift United out of the bottom three.
12.10pmAcross the Slaven Bilic and Moyes regimes, West Ham are eight league games without victory - with the only success in that run an EFL comeback win over Spurs. The last time the Hammers went winless in nine league matches was in May 2011, when they were relegated from the top flight.
12.12pmMoyes did not arrive to much fanfare but what he was expected to do was make West Ham more organised and difficult to beat. It will be a concern then that the division's worst defence has conceded nine goals in Moyes's four league games and are yet to keep a clean sheet.
12.14pmWorryingly, as well as conceding plenty the goals have dried up at the attacking end for West Ham. During their eight-game winless run, the Irons have scored only six times, although this barren spell has largely coincided with Hernandez's absence and he returns today.
12.16pmSTAT: David Moyes, look away now! This is the 41st consecutive Premier League matchday that a side managed by Moyes will start out in the relegation zone, breaking John Gorman's record set at Swindon Town in 1993-94.
12.18pmWest Ham's tally of 10 points after 15 games is their lowest at this stage of a Premier League season. However, it is only three fewer than they had at this stage last term, when they went on to finish very comfortably in 11th in the table.
12.20pmThe odds seem stacked against West Ham today, but they do have a derby specialist in their midst. Nine of Manuel Lanzini's 15 Premier League goals for West Ham have come in London derby matches, including two in as many games against Chelsea.
12.22pmThe Hammers welcome Chelsea on the back of their most encouraging performance yet under Moyes in defeat to runaway leaders Man City. It will be interesting to see if they continue with the three-man defence they used at the Etihad, or revert to a 4-2-3-1. The flexibility of Aaron Cresswell and Arthur Masuaku down the left gives Moyes the option of switching things up easily.
12.24pmHEAD TO HEAD: Just over five minutes until kickoff at the London Stadium, where a victory for Chelsea would be their 50th against West Ham in all competitions. The Hammers have won just two and lost 16 of their last 22 Premier League games against the Blues.
12.26pmPREDICTION: Chelsea won 2-1 in both league meetings last season, while West Ham came out on top by the same score in a League Cup tie at the London Stadium, and I think the 2-1 trend will continue with Conte's men edging out Moyes's boys.
12.28pmSTAT: One final glimmer of hope for West Ham fans - Moyes has won four of his last six home games against the reigning Premier League champions, including three times against Italian managers - Carlo Ancelotti, Roberto Mancini and Claudio Ranieri. Will Conte become number four?
1 minKICKOFF! The action is underway, and it does appear as though West Ham are matching Chelsea and sticking with the 3-5-2 formation from last week, with Cresswell at left centre-back.
3 minGood start from West Ham who have pushed Chelsea back and won a couple of set pieces which have given them the opportunity to load the box. The early deliveries have been a bit sloppy, however.
5 minThe Hammers nearly break on the counter-attack with Antonio down the right. Arnautovic is up with him but Kante recovers as only he can to get a tackle in and allow Chelsea to regroup.
6 minGOAL! WEST HAM 1-0 CHELSEA (MARKO ARNAUTOVIC)
6 minWest Ham's early adventure is rewarded with a sixth-minute goal and finally Marko Arnautovic delivers in claret and blue! It is a lovely move from the hosts as the Austrian plays a one-two with Lanzini, spinning away from Christensen, and places a left-footed shot into the bottom corner.
8 minYELLOW! Arnautovic was understandably pumped with his first West Ham goal, but did get a little overexcited and was cautioned for celebrating in the crowd. What a silly rule that is, though.
10 minWest Ham continue to come forward and nearly catch Chelsea out on the break. Antonio has Arnautovic to his left but his pass is picked off by Cahill and the visitors survive a nervy moment.
12 minThe Irons knock a free kick deep to the back post and Ogbonna, who scored with a header against Man City last weekend, wins the initial header but Christensen clears out of a congested box.
14 minMorata throws himself to ground under the challenge of Reid in the search for a penalty, but there was very little contact and certainly not enough to take down a forward of the Spaniard's stature.
16 minA lot was made of Adrian's inclusion ahead of Joe Hart today and the Spaniard has had a very quiet opening quarter of an hour. He has yet to be tested by Chelsea as West Ham continue to look the more threatening side going forward. Hazard has barely had a touch so far.
18 minYELLOW! Adrian's first significant involvement is to take a catch under pressure from Morata, and then take a knee in the gut from Alonso on the follow through. The Chelsea wing-back is shown a yellow card for his hanging leg by Anthony Taylor, which did seem a little harsh.
20 minCHANCE! First half opportunity for Chelsea as an Ogbonna header falls to Hazard, who screws a left-footed volley across the face of goal and wide. It was a difficult chance but the type that top-class players would make much more of.
22 minWest Ham have just sat off in the last few minutes and Chelsea have managed to grab a foothold in the game. Fabregas is seeing more of the ball and the Blues are starting to move it quicker as a result.
24 minFabregas slips a good pass into the feet of Bakayoko, who is back in the Premier League starting lineup for the first time in four matches. The former Monaco man's shot is blocked by Ogbonna.
26 minThe back-five formation of West Ham is working so far as neither Alonso or Zappacosta have managed to get to the byline yet, with Masuaku and Zabaleta sitting deep to protect this early lead.
28 minSHOT! Kante has some space on the edge of the box and tries to bend a shot into the bottom corner. There is not much power on the effort but it still requires a diving save from Adrian to keep it from threatening the goal.
29 minSAVE! Adrian is called into action again and this time needs to be much sharper to keep out a left-footed swinger from Zappacosta. The rebound falls to Morata who is flagged offside.
31 minWest Ham need to try to get hold of the ball again because, with the amount of possession that Chelsea are having, it won't be long until they find an equaliser. All the Hammers are doing right now is clearing towards Antonio, who is doing his best to hold the ball up against three centre-backs.
33 minMorata has proven to be lethal in the air in his short time in English football, but Reid and Ogbonna do enough to make a header from Zappacosta difficult direct, and it drifts harmlessly wide in the end.
35 minGood defending again from West Ham as Masuaku comes across to thwart Hazard after the Belgian had ran onto a delicate Fabregas chip over the top. Morata beats Reid to the second ball but pokes it way over the crossbar.
37 minMasuaku shows his pace to beat Zappacosta to a 50-50 challenge, although the Italian's weak effort made it more of a 90-10. The Hammers man strides forward with purpose only to drag a shot wide from 25 yards out.
39 minWhen West Ham can get the ball forward, the Arnautovic-Antonio combination is causing problems for Chelsea. The pair link up nicely again but the Austrian just strays offside down the left wing.
41 minMasuaku looks to have Zappacosta's number down the Chelsea right and he helps West Ham win a couple of corners in quick succession. The hosts are finishing the half as well as they started it.
43 minAn incessant Zabaleta press from his own half right up to Courtois lifts the crowd as half time approaches. Getting to the break with a 1-0 lead will be a great checkpoint for Moyes and his players.
45 minArnautovic pins himself up against Christensen and gets a shot away on the turn, but it is a difficult technique and the lack of power on the 20 yard effort makes it easy pickings for Courtois.
1.22pmThat is probably as good a half as Premier League football as West Ham fans have seen at the London Stadium this season, and finally they have some return on their £25m investment in Marko Arnautovic, whose goal gives them a narrow lead at the interval.
1.24pmHowever, the Hammers have been in this position before this season and not gone on to close out victory. Three times they have led at half time and they have not won any of those matches.
1.26pmYou would expect Chelsea to improve in the second half because West Ham cannot keep this intensity up for the whole 90 minutes. So far they have kept Hazard and Fabregas quiet but can that last?
1.28pmArnautovic has been buzzing off the back of his first narrow goal, while Antonio and Masuaku have also been excellent for West Ham to ensure they retain an attacking threat while protecting this lead. Considering their defensive record, the Hammers are likely to need a second after the break.
1.30pmBENCH WATCH: Zappacosta has been completely ineffective down the Chelsea right, and also exposed defensively by Masuaku a couple of times. Conte has Moses in reserve and 45 minutes should not be too much for the former Hammers loanee after his recent return.
1.32pmSUBSTITUTION: News of a change coming from the Chelsea camp, with the extra goal threat of Pedro coming on to replace Bakayoko, which leaves Fabregas in a vulnerable deeper role. The second half is imminent...
46 minKICKOFF: West Ham resume the action as they look to secure a first win under David Moyes...
48 minFabregas looks for Zappacosta in behind but Cresswell comes across to shield the ball behind for a goal kick. The former Ipswich man has slotted in nicely in his new left centre-back position.
50 minCrucial intervention again from Cresswell as he hooks the ball clear from his own six-yard box after Fabregas had squeezed a cross in from the left byline. I think it was Pedro who was waiting for a simple tap-in.
51 minGreat opening for West Ham again as Antonio and Arnautovic break two on three against the Chelsea defence. The Austrian is played through on goal, but the pass came too late from his teammate and the offside flag goes up as Courtois makes a strong save anyway.
53 minZappacosta's cross is poor but Fabregas makes something of it with a diving header which is blocked by Ogbonna.
54 minPENALTY APPEAL! West Ham want a penalty for handball against Christensen. It did definitely strike the Dane's hand, but there was no intent from the Chelsea defender, who could not see the ball when Arnautovic knocked it round him on the left edge of the box.
55 minSUBSTITUTION: We do now see Moses, but surprisingly it is Alonso who is replaced and not Zappacosta. Alonso is a set piece threat with his head and feet and provides more balance, but obviously Conte sees something I don't!
56 minMorata wins a free kick from Obiang 30 yards out, and Cresswell boots the ball at the grounded Spaniard for good measure. Imagine the ruckus that would have followed if that would have been Diego Costa! Morata isn't interested in the theatrics, and Cresswell is lucky to escape a yellow card.
58 minCLOSE! Zappacosta cuts in from the left and tries to bend a shot beyond Adrian, who has his post covered as the ball drifts a few yards wide. He is then booked for time wasting from the subsequent goal kick.
60 minZappacosta and Zabaleta get into a right tussle over the ball in the corner of the Chelsea half, and the West Ham man eventually pushes it too far and gives away a free kick. The Argentine is pumped for this one though and Moyes is lucky to have inherited such a character in this squad.
62 minA shot from Azpilicueta takes a deflection and nearly falls to Morata on the penalty spot, but Adrian is quickly off his line to collect the loose ball. He has not put a foot wrong after being given the nod over Joe Hart, the West Ham goalkeeper.
64 minSUBSTITUTION: Final change for Chelsea as Zappacosta is withdrawn for Willian. That is likely to see Chelsea switch to four at the back, with Azpilicueta moving over to the left. Conte is never going to die wondering and has made all of his substitutions early here.
66 minI wonder when Moyes will turn to his bench for the first time. Antonio has put in a massive shift but looks exhausted and Ayew or Sakho might soon be called upon to do the hard yards up front.
68 minYELLOWS! Reid and Cresswell both enter the book in quick succession with Chelsea upping the ante. The Cresswell offence presents the visitors with a free kick by the right byline from which Reid makes a crucial block from Hazard.
70 minSUBSTITUTION: Sakho is indeed called upon to share the burden with Antonio as Arnautovic leaves to a warm ovation. Could his first West Ham goal end up being a winner?
72 minYELLOW! Obiang has recent become the fifth West Ham player to be shown a yellow card, and the third in the last five minutes. The Hammers need to keep their cool here because a sending off would make seeing out this result very difficult indeed.
75 minHazard runs at Obiang into the box and the West Ham midfielder does well to stay on his feet and limit the Belgian to a corner. Chelsea need some magic from their best player now.
78 minSUBSTITUTION: Antonio has run himself into the ground, quite literally, as the exhausted Hammers forward lies down for a rest forcing Moyes to call on Ayew for the final 12 minutes.
80 minWest Ham have 10 minutes plus stoppage time to hang onto what would be a huge and surprising three points, and a first victory under Moyes. What a morale booster this would be for the Hammers, and what a fatal blow it would be to Chelsea's hopes of defending the title.
82 minKante plays the ball into the feet of Morata and carries on his run, but Ogbonna stands strong in defence to fairly block the Frenchman off.
83 minCHANCE! Glorious opportunity for Chelsea to equalise as Masuaku plays Morata onside only for the Spaniard to fire his shot from 12 yards into the side-netting. That was the chance that the striker has been waiting for all afternoon.
85 minSHOT! Suddenly West Ham appear a little stretched and Hazard darts into some space to create room for a left-footed shot which was always rising over the crossbar. Here comes the Chelsea push, but is it too late?
87 minReid steps up to leave Morata offside from Chelsea's latest attack. West Ham's defence has rightly been panned for much of this season, but he and Ogbonna have been superb all game.
88 minYELLOW! Masuaku is booked for blatant time wasting over a throw-in, moments after Hammers assistant boss Stuart Pearce was sent back to the bench for booting the ball into the stands with the same purpose!
90 minWest Ham are no longer carrying any counter-attacking threat or able to hold the ball up. It is all Chelsea and the Hammers are trying to hold on while the Blues struggle to create something clear cut.
90+1 minThe signal of a minimum of FIVE minutes stoppage time is met with groans around the London Stadium. It would be a hammer blow for the home side to concede now - no pun intended.
90+3 minA clearance falls to Fabregas whose wild shot flies high into the stands.
90+4 minAyew and Masuaku double up against Moses and boot the ball clear. Sixty more seconds for West Ham to hang on.
2.28pmIt is a massive result for David Moyes and West Ham as they get their first win under their new manager. They are still in the relegation zone, but now surely heading in the right direction. As for Chelsea, they miss the chance to close the gap on the Manchester clubs ahead of tomorrow's derby.
2.30pmThat's it from this all-capital clash - (read match report here) - but our Premier League coverage for the weekend is only just beginning. Stay with us for today's 3pm kickoffs, and make sure you return for the huge instalments of the derby triple header tomorrow. Until then, thanks for joining us!