The Reds dominated the first half and led 2-0 at half-time thanks to goals from Virgil van Dijk and Alex Oxlade-Chamberlain.
Odsonne Edouard opened the game up on 55 minutes and Palace threatened an equaliser until a controversial penalty on 89 minutes by Fabinho clinched the points.
City’s winning run came to an end at 13 games on Saturday as they were surprisingly held to a 1-1 draw by Southampton.
Kyle Walker-Peters scored a stunning opening goal after seven minutes but the Saints were ultimately unable to hold on, with Man City levelling midway through the second half via Aymeric Laporte.
Chelsea hosted Tottenham in the big top-four clash on Sunday. After a controversial disallowed goal from Harry Kane, Chelsea broke the deadlock in the second half with a brilliant curler from Hakim Ziyech.
Thiago Silva headed home a free kick to make it 2-0 to Chelsea, and they saw the game out comfortably thereafter.
The Gunners are still yet to win in 2022.
Manchester United's own Champions League hopes were boosted as they defeated fellow challengers West Ham 1-0 at Old Trafford on Saturday.
The Red Devils have been no strangers to last-minute winners from Cristiano Ronaldo this season, but on this occasion it was Marcus Rashford who popped up to seal the victory and push Ralf Rangnick’s side into the Champions League spots.
United had the better of what had gone before and deserved their success, with Rangnick taking credit for the changes he made throughout the game as they made it back-to-back victories in the space of four days.
West Ham, meanwhile, have dropped out of the top four.
At the foot of the standings, meanwhile, there were crucial victories around the relegation zone. Norwich thumped Watford 3-0 away from home on Friday, putting the pressure on Newcastle to get a positive result at Leeds.
Eddie Howe’s men responded with a 1-0 success. Jonjo Shelvey’s free kick was the difference at Elland Road.
Leicester and Brighton drew 1-1 in Sunday's other match.
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