Samir Nasri put them on course for victory in their vital Group D game in Amsterdam with an early strike but it came much against the run of play.
Ajax responded emphatically to get their first win of the competition - and leave 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 Barclays Premier League champions bottom of the group and staring at the possibility of failing to reach the knockout phase for a second successive season.
Source: PA