Manuel Pellegrini's favourites were trailing to Fabio Borini's 10th minute strike for Sunderland and a shock looked on the cards until Yaya Toure and Samir Nasri intervened.
Jesus Navas completed the turnaround late on.
Toure curled in City's equaliser after 55 minutes before Nasri thumped home the winner to snatch the season's first silverware for Pellegrini's title-chasing side.
(Reporting by Martyn Herman; editing by Justin Palmer)
