Cenk Tosun fired the Eagles in front against the run of play in the 39th minute, but Sergio Aguero replied for the Sky Blues in the 82nd and 87th minutes, taking his tally for City to 251 goals.
However, Palace equalised in the final minute of normal time when Fernandinho turned home Wilfried Zaha's cross for an own goal.
For Palace, the stalemate was their fourth consecutive draw in the league.
Kevin De Bruyne and Aguero both came close to breaking the deadlock early on. The former, after missing the target with a first-time effort, rattled the crossbar with a superbly hit free-kick, before the latter forced a good save from Vicente Guaita.
City were perhaps fortunate to escape a penalty when Zaha went down under a challenge from John Stones, but the England international was cleared following a VAR check.
The Athletic's Sam Lee thought Palace should have been given a spot-kick:
After Raheem Sterling headed over, the Sky Blues were made to pay for failing to take their chances when the Eagles converted their first shot on target of the game.
Gary Cahill headed James McArthur's corner into the path of Tosun—on loan from Everton for the remainder of the season—and the striker made no mistake in nodding in from close range.
City should have equalised shortly after the break when Sterling fired wide after being picked out by David Silva, before Guaita made a fine save to deny Bernardo Silva after Cahill had blocked an Aguero shot.
At the other end, the hosts had Ederson to thank when he kept out a Tosun half-volley following a rare counter-attack.
The Manchester side were awarded a penalty after a Jairo Riedewald handball, but the decision was correctly overturned by VAR—the ball had hit his arm after flicking up off his leg.
The hosts finally pulled level when Aguero volleyed home a cross from substitute Gabriel Jesus.
Minutes later, the Argentinian bagged a second when he met Benjamin Mendy's cross with his head.
The Citizens were unable to see out the game, though. After beating Stones in a one-on-one, Zaha put a dangerous ball into the six-yard box that went in off City's Fernandinho.
City travel to Sheffield United on Tuesday, while Palace host Southampton the same day.