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The Premier League champions recovered from two goals down to earn a point and preserve their 18-match unbeaten European home record through Sergio Aguero's second half equaliser.
Ajax captain Siem de Jong scored twice in the opening 17 minutes to stun the hosts but a quick response from City, courtesy of a superb piece of finishing by Yaya Toure after 22 minutes, at least breathed life into the match.
Relentless pressure from Roberto Mancini's team was rewarded when Aguero levelled after 73 minutes but, with only two points from their opening four games, City need every result to go their way in the remaining group matches to advance.
To cap an unrewarding evening for the hosts, their volatile forward Mario Balotelli had to be forcibly prevented from approaching Danish referee Peter Rasmussen at the final whistle after he turned down a late penalty claim.
Replays suggested the Italian had a just claim, after he was tugged in a goalmouth melee, and the unsuccessful appeal may have major repercussions, not only for City's prospects in the competition but Mancini's future as manager.
With Borussia Dortmund drawing 2-2 at Real Madrid in the other match in the section, a City victory would have left them handily placed with a chance of securing a top-two finish.
Instead they remain bottom of the group after four of their six group games behind leaders Dortmund who have eight points, Madrid on seven and Ajax with four.
City paid the penalty for appalling defending in conceding the goals that put Ajax firmly in control.
After nine minutes, Ryan Babel sent over a dangerous corner which Niklas Moisander turned towards goal. City had left de Jong completely unmarked at the far post and the midfielder converted clinically with Joe Hart's goal at his mercy.
Mancini, who had several defensive absentees through injury, must have been even less impressed that his team then conceded an equally poor goal from another corner seven minutes later.
This time midfielder Yaya Toure was at fault by failing to follow de Jong as he ran towards the near post to meet Christian Eriksen's corner and finish with a powerful, angled header.
The terrible start left City facing elimination from the group stage for a second consecutive season.
However, Toure quickly made amends with a magnificent piece of finishing to revive City's fading European aspirations.
Fellow midfielder Samir Nasri, ineffective for most of the evening, sent over a cross from the right and Toure backed skilfully into his marker before scoring with a crisp volley on the turn from six yards.
A header from City defender Pablo Zabaleta might have brought an equaliser before the interval while de Jong could have claimed an incredible first-half hat-trick but his long shot flew just off target.
The introduction of Balotelli at the interval, and fellow stiker Edin Dzeko midway through the second half, signalled Mancini's attacking intent.
City's defence may have been brutally exposed at times this season - particularly in Europe - but their attacking arsenal boasts talent on a par with most teams.
Hart was still required to make solid stops, from de Jong and a Lasse Schone free kick, before the almost inevitable equaliser came courtesy of Aguero.
The Argentina forward raced onto Balotelli's flick-on from Hart's long kick and outpaced Toby Alderweireld before scoring, right-footed, from a step inside the penalty area.
Balotelli might have netted with a fine header from a Nasri free kick, which was well saved by Kenneth Vermeer, and Aguero had the ball in the net once more only for his effort to be harshly ruled out for offside against Aleksandar Kolarov.
Then came the late drama, which included another superb save from Hart to deny Eriksen.