Cardiff City were denied a winning start to the post-Malky Mackay era after Sunderland snatched a point away from home deep into stoppage-time.
Hull City delivered a scintillating second-half display to thrash Fulham at home, while West Ham United remain in the bottom three after drawing with West Bromwich Albion.
Aston Villa ended a run of four consecutive defeats in the Premier League after a home draw with Swansea City.
Edin Dzeko scored a second-half winner as a much-changed Manchester City were forced to work hard to beat lowly Crystal Palace 1-0 at the Etihad Stadium.
City pressed for much of the match in search of a goal, but also had to be vigilant at the back, with Palace starting to ask questions of Joe Hart in the second half.
However, Manuel Pellegrini's side made the breakthrough on 66 minutes when Palace could only partially clear a cross into the box and Jesus Navas teed up Dzeko to side-foot emphatically home.
Palace responded immediately and almost scored an equaliser when Joel Ward's 25-yard shot produced a flying save from Hart to tip the ball round the post.
Manchester United made it six wins in a row in all competitions after half-time substitute Danny Welbeck's goal secured a 1-0 victory at Norwich City.
United grabbed the lead decisively on 57 minutes when Ryan Bennett's clearance clattered off Danny Welbeck and then ricocheted back to the England international off team-mate Javier Hernandez. Welbeck was left clean through and he took the ball round John Ruddy before sliding his shot into an empty net.
Sunderland produced a stunning late comeback to deny Cardiff victory in the first game since Malky Mackay's sacking following a 2-2 draw in south Wales.
Cardiff dismissed Mackay on Friday and enjoyed a flying start against fellow strugglers Sunderland when Jordon Mutch's shot deflected off Modibo Diakite on six minutes and flew past Vito Mannone.
Mutch, who was booked for diving in the first minute, then turned provider on 58 minutes when his low cross was converted by former Sunderland striker Fraizer Campbell.
The Black Cats set up a thrilling finish on 83 minutes when half-time substitute Steven Fletcher escaped his marker and stuck out a leg to turn home Emanuele Giaccherini's cross.
And, in the fifth minute of added time, the ball was teed up for Jack Colback just inside the box and the midfielder's shot took a decisive deflection to earn Sunderland a point.
Hull City midfielder Tom Huddlestone can finally get his hair cut after scoring his first goal since April 2011 in a memorable 6-0 hammering of Fulham at the KC Stadium.
After a scoreless first half, Hull went ahead on 49 minutes when Huddlestone's corner dropped to Ahmed Elmohamady and he took a touch before volleying inside the post.
Hull doubled their lead on the hour mark when Yannick Sagbo played in a dangerous first-time cross and Robert Koren stretched out a leg to divert the ball past David Stockdale.
A third goal arrived just three minutes later when Jake Livermore fed a pass into George Boyd, who turned on the edge of the area and angled his low shot beyond Stockdale.
Midway through the second half, Huddlestone finally ended his long goal drought, and wait for a trip to the barbers, as he feinted on the edge of the box before driving a low shot home.
Huddlestone came close to a second goal on 74 minutes when his 40-yard free-kick came back off the crossbar, but substitute Matty Fryatt was on hand to volley home the rebound.
Koren notched his second goal of the game with six minutes to go when the midfielder swept home the loose ball after Stockdale could only parry Danny Graham's shot.
West Ham were held to a 3-3 draw by West Brom in a topsy-turvy encounter at Upton Park in a game which Sam Allardyce had described as 'must win' in the build-up.
The home side took the lead on four minutes when Carlton Cole flicked on a long punt from Jussi Jaaskelainen to Joe Cole, who played a neat one-two with Kevin Nolan before firing a low right-footed shot past Ben Foster.
West Brom equalised five minutes before half-time when Christ Brunt played a sublime through ball to Nicolas Anelka and the Frenchman dinked his shot over Jaaskelainen.
Albion then took the lead in controversial circumstances on 45 minutes after winning a dubious corner from a Joe Cole challenge with James Morrison. Brunt's corner was headed goalbound by Jonas Olsson and, after Guy Demel blocked tamely on the goalline, Anelka reacted first to poke the ball into the back of the net.
Allardyce sent on Modibo Maiga for Carlton Cole at the break and the substitute equalised on 65 minutes when his curling left-footed effort from outside the box skipped over the diving Foster's right hand.
Hammers fans were in full voice again two minutes later as Maiga's knock-down from a Mark Noble cross found Nolan, who volleyed home from close range to make it 3-2.
But it was 3-3 on 69 minutes when Saido Berahino cut onto his left foot in the 18-yard box and steered an angled drive into the bottom corner.
Out-of-form duo Aston Villa and Swansea settled for a 1-1 draw at Villa Park after Gabriel Agbonlahor's early goal for the hosts was cancelled out by Roland Lamah.
Villa went ahead on seven minutes with their maiden first-half goal at home this season in the Premier League after Andreas Weimann slipped in Agbonlahor to finish confidently.
However, Swansea equalised on 36 minutes when Pablo Hernandez crossed from the left of the penalty area and Lamah squeezed his header between the post and scampering Villa keeper Brad Guzan to leave Michael Laudrup's side without a win in five league games.