The Blues bounced back from their defeat at Sunderland on Boxing Day to record a third win in the last four and keep leaders Manchester United within striking distance.
City got off to the best possible start at Carrow Road, with a quickfire brace from recalled Bosnian striker Edin Dzeko which put them 2-0 up inside four minutes.
A deflected free-kick from Anthony Pilkington halved the deficit before midfielder Samir Nasri was shown a red card for pushing his head into Norwich defender Sebastien Bassong as the pair squared up on the far touchline.
City, though, did not buckle and extended their lead at the start of the second half with a well-taken effort from Sergio Aguero and then twice withstood Norwich's threatened comeback to close out the result.
"To remain with 10 players here is not easy but I was sure we could still score goals and we had our chances," said Mancini, who confirmed City are likely to appeal Nasri's red card which carries a three-match suspension.
"I am happy because we continued to play and did not just sit deep with our players.
"We have won two games now when we had only 10 players, also at West Brom, and that should say we are a strong team.
"The season is long and we still have a chance."