Mancini’s side can still mathematically qualify for the knockout stages but it is a long shot after the Premier League club failed to beat the Group D underdogs at home.
Ajax captain Siem de Jong took advantage of shocking City defending at corners to put the visitors 2-0 up after 17 minutes.
Yaya Toure reduced the deficit soon afterwards with an acrobatic volley to give his team hope it could still be in the competition by the time it hosts Real Madrid in three weeks.
Sergio Aguero scored the equalizer with 16 minutes left but Ajax defended stoutly to keep out a late City charge.
The draw leaves City rooted to the bottom of the toughest of the eight Champions League groups, with just two points, five adrift of second-placed Borussia Dortmund.
Mancini's team teeters on the absolute precipice, requiring victories from the final two matches - home to Real and away to Dortmund - and other results to go its way to be involved in the competition in February.
For this must-win fixture, Man City made three changes from the team that drew 0-0 at West Ham last Saturday, with Aguero, Pablo Zabaleta and the fit-again Javi Garcia replacing Mario Balotelli, Edin Dzeko and the injured Kolo Toure.
With City experiencing a glut of defensive injuries, there was one unfamiliar face on the substitute’s bench, 18-year-old center back Courtney Meppen-Walter, a member of City’s development squad.
The rookie could hardly have defended worse than his more senior colleagues early on as Ajax seized upon some dreadful marking to score twice from corners.
After nine minutes, Matija Nastasic demonstrated how raw he is at this level by, first, making a horrible hash of his clearance to concede a corner from a low cross and, then, failing to deal with the resulting set piece, prodding the ball back across goal to allow De Jong to loft in the opener.
Eight minutes later, alarmingly lax marking from City allowed De Jong to steal in to meet a near-post inswinger and guide his free header past Joe Hart.
Ajax may not have sold out the away end but the visiting fans were by now bouncing up and down in coordinated jollity as the Group D underdogs scented a famous win and an improbable passage into the Champions League knockout stages.
Toure pulled one back for City after 22 minutes with a fabulous piece of skill, chesting down a cross at the back post and then volleying past Kenneth Vermeer.
The home side looked dangerous every time it attacked but a brilliant clearance from Ricardo van Rhijn and Zabaleta’s failure to take advantage of a fine headed opportunity meant that Ajax ended an even half with a 2-1 advantage.
Mancini went all-out attack after the break, replacing Garcia with Balotelli and it was a move that paid dividends as the Italian’s flick-on set up Aguero for the equalizer.
Oddly, the Argentine’s footing had let him down on a number of occasions in good positions during the game but he managed to stay on his feet after 74 minutes when Balotelli headed on Hart’s goal-kick to slickly level the scores.
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