Despite taking the lead, the loss means that City have two points from their opening three matches in the competition.
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The hosts had a sight of goal after nine minutes, but Christian Eriksen flashed his half-volley wide of Joe Hart's post.
The visitors nearly went in ahead at the break, but they were undone right on the half-time whistle as Siem de Jong slotted in from Ricardo van Rhijn's cross.
City came out strongly for the second half but found themselves behind on the hour when Niklas Moisander took advantage of some slack defending to head in from a Christian Eriksen corner.
The visiting defence looked vulnerable again 10 minutes later and they were duly punished as Eriksen was allowed to run at goal before firing into the bottom corner.
Roberto Mancini's side nearly pulled a goal back in the 77th minute, but Nasri shot wide after showing some great control in the box.