A brace from Edin Dzeko, a goal from Sergio Aguero and a Mark Bunn own goal secured a thrilling victory for Roberto Mancini's side, despite a Samir Nasri red card just before half time.
Dzeko got City off to a flyer with the Bosnian opening the scoring with just over 60 seconds on the clock, before adding a second just minutes later to give Mancini's side a two goal lead inside the first five minutes.
Norwich then fought back with an Anthony Pilkington free-kick to reduce the deficit to one in a frantic first 15 minutes.
The drama did not stop there as just before half-time Nasri was then giving his marching orders after reacting to a Sebastian Bassong challenge.
Norwich heads did not drop though, as they again got themselves back into it after Russell Martin headed in from six yards to put the pressure back on Mancini's side.
City responded in style with Dzeko firing an effort towards goal which cannoned off the back of Norwich keeper Bunn before rolling over the line, putting us two ahead for the third time.
The Canaries did grab one back again as Martin netted his second, but Mancini's men saw out the final 15 minutes to record a memorable Carrow Road triumph.
Play the video to enjoy the best of the action...