© PA Photos
Siem de Jong silenced the home crowd in the 10th minute, firing into the roof of the net from a tight angle to give the visitors the lead.
He doubled his account for the evening seven minutes later, capitalising on some poor marking from a corner to ghost in and nod past Joe Hart.
Pablo Zabaleta had a chance to level for the home side, but his header was directed straight at the keeper.
Roberto Mancini's men found it a struggle to break down Ajax, who have defended astutely.