City was 2-0 ahead after four minutes at Carrow Road thanks to a double from Edin Dzeko, but Anthony Pilkington's free kick halved the deficit before Nasri was dismissed after squaring up to Sebastian Bassong in the 44th minute.
Sergio Aguero restored City's two-goal lead five minutes after halftime, only for Russell Martin to make it 3-2 in the 63rd and then 4-3 in the 75th after Norwich goalkeeper Mark Bunn had deflected in Dzeko's low drive for an own goal.
The champions had some nervy late moments but hung on to stay within touching distance of United heading into 2013.
After starting the game in such dominant fashion, City _ surprisingly beaten 1-0 by Sunderland on Wednesday _ looked in the mood to cruise to victory, only for the momentum to change because of Nasri's straight red card.
It was a dubious decision by referee Mike Jones, with the France playmaker squaring up to Bassong after being clattered by a robust challenge from the defender. The pair's foreheads clashed _ and Nasri motioned a butt _ but while Bassong was given a yellow card, the City player saw red.
"Both players came forward with their heads, so if you are going to send off one, then you have to send off both, not only Samir,'' City manager Roberto Mancini said.
"Samir said he touched his head, OK, but the other player has done the same . it is incredible.''
Mancini will be delighted to see his strike force answer his call for more goals, having criticized them this week for being "soft'' in recent games.
Dzeko replaced the ill Carlos Tevez up front to score twice and almost grab a hat trick, with City's fourth credited as an own goal after his shot hit the post, came back out and rebounded off Bunn into the back of the net.
"The guys played very well,'' Mancini said. ``It was like at Sunderland but this time we scored. Today we showed we are there for the title.''
Dzeko's first came in the second minute when he sidefooted in coolly from 12 yards and the Bosnia striker made it eight for the season by slotting into an empty net from Aguero's cut-back.
City was left aggrieved when Pilkington's free kick deflected in off Gael Clichy, a double blow since the original foul on Vincent Kompany was harshly awarded.
Down to 10 men for the second half, City needed a quick lift and Aguero provided it, running onto Yaya Toure's fine through-ball in the 50th and scooping a delightful finish over Bunn.
Martin scrambled in close range goals either side of Bunn's own goal, but Norwich couldn't find the equalizer.
"I am delighted with the result and the performance,'' Mancini said. ``After the sending off it was difficult. However, we fought well.''