Jordi Gomez's first-half penalty and a tap-in by James Perch gave Wigan, who beat City 1-0 in the FA Cup final at Wembley last May, a place in this year's semi-finals against Arsenal.
City, who won the League Cup last Sunday and need to overturn a two-goal deficit to Barcelona in the second leg of their Champions League tie next week, looked jaded against their second-tier opponents until they staged a thrilling late fightback.
Frenchman Samir Nasri made City's breakthrough in the 68th minute with a sweet 20-metre left-foot strike.
Substitute Edin Dzeko came within inches of forcing a replay, but Emmerson Boyce produced a remarkable last-ditch block and Wigan, managed by former City favourite Uwe Rosler, held on.
In the other quarter-finals played on Sunday, Sheffield United beat Charlton Athletic 2-0 and Hull City defeated Sunderland 3-0 to set up the second semi-final.
Arsenal beat Everton 4-1 on Saturday.
Image: James Perch celebrates after scoring goal for Wigan Athletic
Photograph: Jan Kruger/Getty Images
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