The Canaries finish in 11th place after a close victory over last season's champion, with the home side unable to cope with the superb attacking play.
The Canaries took the lead through Anthony Pilkington, although they weren’t ahead for long as Jack Rodwell equalized three minutes later for City.
Norwich went ahead again thanks to Grant Holt, and despite another Rodwell equalizer, the Canaries were not to be denied as Jonny Howson hit a superb winner.
Manchester City made five changes for this game, with Edin Dzeko leading the line up front as Sergio Aguero was only fit enough for the bench.
Meanwhile, Norwich replaced Sebastian Bassong with Steven Whittaker in what was the only change from the 4-0 victory over West Brom.
Both sides started brightly in the opening exchanges, with Grant Holt proving to be a challenge for the home defenders, turning and shooting wide after showing good strength.
City also started well, as Dzeko had a header well held by John Ruddy after a superb cross from Samir Nasri.
Norwich created the best chance of the half midway through. After a good corner from Snodgrass, Ryan Bennett saw a header cleared off the line by Joleon Lescott.
The Canaries took the lead minutes later. Some shambolic defending by the home side left a chance and Pilkington was on hand to sweep home a pass from Wes Hoolahan.
Manchester City was behind for just three minutes, however, as Dzeko laid off a Nasri pass into the path of Rodwell, who slotted the ball home well.
The home side exerted a good deal of pressure on Norwich’s goal but even the efforts of the workmanlike Carlos Tevez could not add another and the clubs went in level at halftime.
After Tevez was replaced to a standing ovation early in the second period, the Canaries continued to exert some pressure and had an appeal for a penalty turned down.
This pressure eventually told for the as Snodgrass played a delightful square ball into the area and Holt was on hand at the back post to finish with aplomb.
Once again, City found an equalizer within minutes. Rodwell accelerated away from Whittaker from inside his own half and finished low past Ruddy.
The Canaries bounced back again minutes later, as Howson rampaged forward from his own half and finished calmly under no pressure whatsoever to give his side the lead again.
It took City until the 79th minute to create another genuine chance, as James Milner played a low cross that Dzeko turned just wide at the near post, but the club could not find another equalizer, with Norwich taking a 3-2 win to finish the season in 11th place.