Roberto Mancini's men raced into a two-goal lead within four minutes through Bosnian striker Dzeko before Anthony Pilkington's deflected free-kick halved the deficit.
A superb first half ended in controversy when Samir Nasri saw red for his angry reaction to a Sebastien Bassong tackle that saw the Norwich defender booked.
The champions extended their advantage once more five minutes into the second half through a cool finish from Aguero, only for Russell Martin to head home from close range just past the hour.
The visitors made it 4-2 just four minutes later when Dzeko's shot from the angle rebounded in off goalkeeper Mark Bunn before Martin's second goal 15 minutes from time ensured a nervy finish.
However, the champions held on for what could be a vital victory and condemned Norwich to a third successive league defeat.
City, who were looking to bounce back following a Boxing Day defeat at Sunderland, were ahead inside two minutes. Aguero and David Silva combining to set up an unmarked Dzeko to sweep home from six yards before Aguero presented Dzeko with a tap-in just two minutes later.
There was some brief respite for Norwich 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.
Things got ugly 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.
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 who produced an exquisite chip to beat the advancing keeper and put City back in control.
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 Dzeko sprung the offside trap and burst into the Norwich penalty area, his shot beating 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 and Martin swept the ball in again from point-blank range.