Eze put Palace ahead with their first attack of the match on Saturday before an inspired Dean Henderson saved Omar Marmoush's penalty.
But Henderson could count himself fortunate to still be on the pitch after he earlier appeared to deny Erling Haaland a scoring opportunity when he handled the ball outside his area in the game's flashpoint.
Daniel Munoz's strike after the interval for Palace was then chalked off by VAR with Oliver Glasner's side having to subsequently thwart wave after wave of City possession to see out the victory and etch their names into the history books by winning the FA Cup for the first time and sealing a place in the Europa League.
For City, manager Pep Guardiola will have to wait for the 19th trophy of his tenure as his side's defeat here caps a bitterly-disappointing campaign which could yet see them miss out on the Champions League next season.
It had started encouragingly for Guardiola's players, who had Palace camped inside their own half for the opening quarter-of-an-hour with Haaland meeting Kevin De Bruyne's cross after six minutes to force a decent save from Henderson.
But suddenly, Palace were ahead. Jean-Philippe Mateta held up a long punt forward before he turned the ball round the corner to Munoz. The Colombia international's pinpoint pass was met by Eze, who steered the ball past Stefan Ortega.
The game turned on the Henderson incident as he handled the ball outside of the area to prevent Haaland from meeting Gvardiol's long ball forward.
With only Henderson to beat, was Haaland denied a clear scoring opportunity? Controversially, VAR did not think so - claiming the ball was going away from the goal - and Henderson was in the clear.
City had every right to feel aggrieved, but they were then awarded a chance to wipe out Palace's lead when Tyrick Mitchell's sliding tackle missed the ball and took out Bernado Silva instead. Yet Henderson pulled off a strong one-handed save to deny Marmoush.
With half-time approaching, the keeper was back in action again when he produced a flying save to stop Jeremy Doku's curling effort.
Wild scenes in the Palace end were curtailed after the break when VAR intervened to rule out Munoz's goal with Sarr in an offside position.
Henderson again pulled off a key save in the dying minutes, twice denying 19-year-old substitute Claudio Echeverri from scoring on his debut.
