The champions, beaten at Sunderland on Boxing Day, were off to a flying start when they went 2-0 up inside four minutes after Edin Dzeko's quick brace.
Norwich reduced the deficit through Anthony Pilkington's deflected free-kick, before Nasri was shown a red card by referee Mike Jones on 44 minutes for putting his head into the face of Sebastien Bassong (pictured right).
Sergio Aguero's neat flick made it 3-1 at the start of the second half, before Russell Martin's header reduced the deficit again.
Dzeko put City back in command at 4-2 after his shot cannoned off the post and in off goalkeeper Mark Bunn, only for Martin's scrambled effort to set up a grandstand finish.
However, Norwich - who had beaten both Arsenal and Manchester United at Carrow Road during a 10-match unbeaten run - could not conjure an equaliser as they suffered a third successive league defeat.
City were ahead inside two minutes.
A long ball down the left was worked back inside by Aguero and David Silva for an unmarked Dzeko to sweep home from six yards
The champions swiftly doubled their lead.
Play was allowed to continue after a crunching tackle from Vincent Kompany on Bradley Johnson.
Aguero was away again, darting into the right side of the Norwich penalty area before drawing the goalkeeper and then cutting the ball back to present Dzeko with another tap-in.
The Canaries had been thrashed 6-1 here by City last season, and another such defeat looked on the cards as the Blues continued to press.
There was some brief respite on 13 minutes when Kompany tripped Robert Snodgrass just outside the Manchester City penalty area.
The Scotland international shaped to take the free-kick himself, but instead ran over the ball, as Pilkington drove in a low shot which deflected off Gael Clichy and past Joe Hart.
Bunn then needed to be alert to tip over a spooned clearance from former City full-back Javier Garrido which was dropping into the Norwich net following another lighting break from the visitors.
City remained a threat, as Yaya Toure flashed a 20-yard drive just wide following more patient build-up from the champions.
Things got ugly again just before the break when Bassong upended Nasri on the far touchline.
The duo squared up face-to-face, with Nasri pushing his head into the Norwich defender - which on consultation with the assistant referee resulted in a red card for the France midfielder, and a caution for Bassong.
The decision perplexed City boss Mancini, and left him with some changes to mull over at half-time the rain started to lash down.
Norwich made a substitution just before the interval, with striker Steve Morison hobbling off because of what appeared to be a thigh problem and on-loan Harry Kane replacing him for a first appearance since breaking his metatarsal in September.
City were soon back on the offensive following the restart and restored their two-goal cushion on 50 minutes.
Toure delivered a perfectly-weighted angled ball which Bassong could not cut out and dropped into the path of Aguero. The Argentina forward produced an exquisite chip to beat the advancing keeper and put City back in control.
To their credit, the home side continued to look to get back into the match, as a low shot from substitute Jonny Howson's was touched wide by Hart.
From the resulting short-corner, it was 3-2 when Bassong's header back across the six-yard box from Snodgrass' clever centre was guided into the top right corner by Martin on 63 minutes.
However, City quickly restored their advantage as the broke away down the right.
Dzeko sprung the offside trap to burst into the Norwich penalty area to beat Bunn at the near post, the ball bouncing off the woodwork, then the keeper's back and into the net.
A quite amazing match sparked into life once more with 15 minutes left when another Norwich corner caused panic in the City six-yard box as Hart could only half punch clear before Martin swept the ball in again from point-blank range.
Mancini looked to shore things up when he replaced Dzeko with defender Joleon Lescott.
Bunn redeemed himself with a brilliant one-handed save from Aguero at point-blank range.
Despite five minutes of stoppage time, there was to be no more drama as City closed out what could prove to be a crucial victory.Tuesday 1st January 2013 West Brom V Fulham Man City V Stoke Swansea V Aston Villa Tottenham V Reading West Ham V Norwich Wigan V Man Utd Southampton V Arsenal Wednesday 2nd January 2013 Chelsea V QPR Liverpool V Sunderland Newcastle V Everton
Source: Manchester City MAD