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Roberto Mancini's men got off to a bad start, as Siem de Jong latched onto a knockdown inside the area to give the visitors the lead in the 10th minute.
The striker finished smartly, lifting the ball into the roof of the net from a tight angle to evade Joe Hart.
De Jong made it 2-0 to Ajax seven minutes later, as the City defence stood static from a corner while he stole in and headed home.
Yaya Toure cut the deficit back to one when he volleyed home in the 22nd minute, before Pablo Zabaleta missed a headed chance that would have seen them into the half-time break on level terms.
Sergio Aguero had the chance to restore parity to the scoreline midway through the second half. The Argentine rounded Kenneth Vermeer, but slipped at the vital moment and could not convert the opportunity.
The home side enjoyed more of the ball in the second half, without really creating much. However, Aguero pounced on Mario Balotelli's knockdown to give City genuine hope of three points.
Aguero thought that he had won it for the hosts with two minutes remaining when he stuck a leg out at the near post the steer the ball past Vermeer, but the linesman had flagged Alexandar Kolarov offside.
There was a touch of controversy at full-time, as the referee blew up following a City corner.
Mancini rushed onto the pitch after the whistle and shouted angrily at Peter Rasmussen, feeling that Mario Balotelli was having his shirt pulled inside the area.