Samir Nasri put them on course for victory in their vital Group D game with an early strike, but it came much against the run of play.
Ajax responded emphatically to get their first win of the competition, leaving City still looking for theirs, with goals from Siem de Jong, Niklas Moisander and, with the aid of a deflection, Christian Eriksen.
The defeat left the Premier League champions bottom of the group and staring at the possibility of failing to reach the knockout phase for a second successive season.
Ajax were supposedly the group's weakest link but looked anything but in a game which, although City had their moments, the hosts dominated.
City boss Roberto Mancini had rewarded Edin Dzeko for his recent form with a starting role alongside Sergio Aguero up front.
Yet it was some time before either of them were able to get into the game as Ajax, with former Liverpool players Ryan Babel and Christian Poulsen prominent, started well.
The hosts dominated possession for large spells although they were surprisingly unable to test goalkeeper Joe Hart enough.
They created their first opportunity after four minutes when De Jong played a one-two with Babel outside the box but then curled his shot over.
De Jong showed more fine control as he chested down a cross and then laid off for Eriksen, but the Danish playmaker was also off-target with his effort.
There were worrying signs for City as Yaya Toure was helped to the sidelines for treatment after a Daley Blind tackle, but the key midfielder was back on, albeit looking in discomfort, two minutes later.
More neat interplay from Ajax led to Eriksen releasing Babel with a sharp pass but City captain Vincent Kompany made a timely challenge.
City finally responded on 22 minutes with a swift and penetrating counter attack leading to them snatching the lead.
Micah Richards raced clear down the right and fed inside to James Milner, who in turn set up Nasri for a clinical first-time finish over the diving Kenneth Vermeer.
Aguero almost doubled the lead moments later but Toby Alderweireld got back to block for a corner.
A clever backheel by Dzeko then released Richards again. The right-back initially hesitated as he looked for a pass but with no decent options he powered on and forced a good save from Vermeer.
Ajax began to assert themselves again as Ricardo van Rhijn found space and cut inside only to shoot over. Lasse Schone also missed the target and Babel fired straight at Hart.
The equaliser came on the stroke of half-time as Van Rhijn delivered a low cross along the edge of the area.
Gareth Barry was unable to cut it out and De Jong met it with a fierce low strike which gave Hart no chance.
City looked to produce a quick response after the interval as Milner charged forward but Alderweireld cut out his cross for Aguero.
Dzeko raced forward as Barry crossed to the near post but the Bosnian could not get a header on goal.
Ajax also showed some urgency as they chased the win and Poulsen almost played Eriksen in on goal but Toure crucially cut out his pass.
The Dutch side claimed the lead from a corner in the 57th minute as Moisander leapt in front of Joleon Lescott and glanced a header across Hart's goal into the bottom corner.
Nasri twice brought Vermeer into action to push away corners as City tried to hit back but Roberto Mancini decided to bring on the extra attacking threat of Aleksandar Kolarov for Lescott.
Kolarov was soon into the action and whipped in a cross but Milner headed wide and Ajax forced forward again.
Babel had a route to goal blocked by Kompany but the hosts sustained the pressure and Eriksen ran across the area and shot past Hart with the aid of a big deflection off Clichy to all but seal victory.
Tobias Sana had a chance to further bury City as he raced on to a fine pass from Eriksen but Hart bravely rushed out to block his shot.
Play went straight to the other end and Dzeko found himself in on goal but Vermeer denied him.
City poured forward in the latter stages but Dzeko, after beating Vermeer, was denied again as Moisander got back.
Carlos Tevez and Mario Balotelli came on to form a four-pronged attack but City were unable to find a way through.
Aguero headed wide and Dzeko could only find Vermeer in fine form with late efforts.