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 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."
City remain seven points behind United, but Mancini is not reading too much into the gap just yet.
"At this moment, United are better than us, because they are on the top. But for us, it is not important to look at the table now. We need to work, win and maybe in February we will look again," The Italian added.
Mancini had freshened things up after their defeat at the Stadium of Light, opting for Dzeko in place of Carlos Tevez at the spearhead of the attack.
Despite walking off with the matchball, Dzeko was denied a hat-trick after his breakaway effort on 67 minutes has been officially credited as an own-goal after the ball bounced off the post and the back of Norwich keeper Mark Bunn before rolling into the net.
Norwich had turned over both Arsenal and Manchester United during their 10-match unbeaten Premier League run, but have now lost three on the spin.
Despite two close-range efforts from Russell Martin - deputising as captain in place of injured striker Grant Holt who remains touch-and-go for the New Year's Day trip to West Ham - the Canaries could not conjure some late drama.
Nevertheless, manager Chris Hughton remains positive for the challenges ahead as his team look to maintain distance between themselves and the relegation dogfight.
He said: "The real plusses are we have played away at West Brom, who had a wonderful season, the Champions League winners (Chelsea) and now the Premier League winners in consecutive games, and have been in all of them, losing by just one goal, so that is a measure of where we are.
"For clubs like ourselves always look at it as a bonus when you get scalps against the big ones, and we were fortunate to have two of those here.
"But we are not unrealistic of how difficult it is, and the quality these sides have.
"Our season will not be defined on these matches, it will be about making sure we get enough points when we can elsewhere."