The Premier League champions were looking to establish themselves alongside Europe's elite, but after they had earned just one point from their two opening fixtures against Real Madrid and Borussia Dortmund, the odds were already stacked against City reaching the knockout stages.
Despite playing at their own ground, Ajax were classed as the underdogs for this contest, but they began the encounter brightly, with Christian Eriksen's 20-yard effort managing to beat Joe Hart but just evading the goalkeeper's post.
The Danish playmaker created a similar opportunity minutes later but he could only produce the same result, and against the run of play, Manchester City stunned their hosts by taking an undeserved lead.
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Micah Richards and Milner combined to begin a City counter-attack, before the ball found Samir Nasri, who wasted no time in striking a first-time effort past the goalkeeper from 16 yards to hand his side the initiative.
All of a sudden, City appeared to be in the ascendancy, with Richards testing Kenneth Vermeer with a powerful attempt, before Nasri's through-ball just failed to find Aguero.
However, just as it appeared as though City would lead at the break, Ajax equalised. Some fine work from Ricardo van Rhijn resulted in the full-back crossing for Siem de Jong, who made no mistake with a low drive into the bottom corner.
Mancini would have demanded an immediate response from his team in the second half, and he nearly got one through Gareth Barry and Yaya Toure, but they couldn't convert when given an opening in and around the Ajax penalty area.
That gave the home side plenty of encouragement and they proceeded to go up the other end to take the lead. After conceding a cheap corner, City's marking was poor from the set piece, and Niklas Moisander took full advantage to rise above the defence to head home from close range.
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It had been a topsy-turvy game in Amsterdam, and that pattern continued as City threatened through Dzeko and Milner, but with just over 20 minutes remaining, Ajax netted their third goal to effectively seal the three points.
Barry lost possession of the ball inside his own half and after Tobias Sana had helped the ball to Eriksen, the Danish international's shot found the back of the net via a deflection off Gael Clichy that deceived Hart.
City soon reverted to a back three and while they looked increasingly likely to halve the deficit, they also looked vulnerable in defence, but neither team could find a fifth goal in the match.
Aguero and Dzeko continued to press for the English outfit as the 90-minute mark came ever closer, but Ajax were dealing with each attack efficiently, and that was enough to help them to a memorable triumph.
This demoralising defeat all-but-condemned Manchester City to an exit from the competition, leaving them to compete with Ajax for a place in the Europa League after Christmas.
However, after they could only record a 2-2 draw with the Dutch side at the Etihad Stadium, a failure to win either of their final two matches resulted in Mancini's men facing the remainder of the campaign without the prospect of any European football.