Nasri off but Dzeko double fires City
Roberto Mancini's side were 2-1 up thanks to two goals from Bosnian striker Dzeko in the first four minutes at Carrow Road when Nasri was shown a straight red card for head-butting Sebastien Bassong just before the break.
Sergio Aguero made it 3-1 and, although Russell Martin deduced the deficit, an own goal conceded by Norwich goalkeeper Mark Bunn made it four before Martin struck again to set up a tense finale.
The win means champions City remain seven points behind United heading into the new year.
City's title hopes suffered a serious setback with a 1-0 defeat at Sunderland on Boxing Day, but they took just 80 seconds to take the lead through Dzeko.
The Norwich defence claimed the Bosnian was offside but the linesman correctly ruled that Martin had played him on.
Referee Mike Jones allowed play to continue and the Belgian's pass found Aguero on the right of the box.
Norwich goalkeeper Bunn came out to challenge but Aguero was able to check and calmly lay the ball back for Dzeko to side-foot into the net despite the best efforts of defenders to close him down.
Kompany was penalised while making a legitimate tackle on Robert Snodgrass just outside the visitors' box - and Norwich pulled one back from the resulting free-kick in the 15th minute.
Bunn then had to pull off an acrobatic tip-over to prevent Javier Garrido, the former Manchester City defender, scoring a bizarre own goal with a clearance that resembled a shot.
An exciting half ended with more controversy as Nasri was sent off, for aiming a head-butt at Bassong. The Norwich defender went in hard on the France international, who reacted angrily.
Nasri pushed his forehead against Bassong's forehead and then flicked it forward before referee Jones consulted with his linesman and produced a red card for Nasri and a yellow for Bassong.
That setback couldn't rattle City and Aguero produced a superb finish to increase their lead in the 50th minute after Yaya Toure's ball over the top had hit Bassong on the back.
Hart had to dive to turn a low drive from substitute Jonny Howson round a post and Norwich made it 3-2 from the corner.
Snodgrass's delivery found Bassong at the back post and his header was headed home by Martin in the 63rd minute.
The champions needed a response and again Dzeko was the man who provided it with the visitors' fourth four minutes later.
It was a close offside call but play continued and the striker saw his shot hit Bunn's near post and rebound off the goalkeeper into the net.
Again Norwich pulled one back with Martin the scorer after Hart and his defenders failed to deal with a corner.
Matija Nastasic did well to block from substitute Harry Kane but Martin was there to divert the ball into the net the moment it came back via Johnson's shot.
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