Ravel Morrison handed the Hammers a first-half lead but Norwich bounced back with three second-half goals.
Gary Hooper scored from the penalty spot to restore parity before a wonderful Robert Snodgrass free-kick handed City the lead and Leroy Fer settled the contest deep in stoppage-time.
Both sides also saw efforts come back off the woodwork but it was Norwich, who had started the day in the bottom three, who held on to take all three points and climb above the Hammers and up to 15th place.
West Ham started to exert their authority on the game with Norwich proving wasteful in possession all too often, looking nervous.
Norwich goalkeeper John Ruddy was forced into the first meaningful save of the game on 12 minutes as he pushed away Kevin Nolan's effort. The ball was eventually turned behind for a corner with Guy Demel's header coming back off the crossbar.
It was Nolan who had the next good chance as the West Ham skipper opened up his body to take Razvan Rat's cross first time, but his shot was too close to Ruddy to cause the England goalkeeper problems.
Nolan was once again heavily involved as the visitors deservedly broke the deadlock through Morrison.
After seeing his initial shot blocked, Nolan collected the ball on the byline and pulled it back perfectly for Morrison to tuck his chance away and give Allardyce's side the lead after a long spell of dominance from the Hammers.
Nathan Redmond was introduced from the bench by Chris Hughton in the immediate aftermath of the goal as Anthony Pilkington appeared to have been struggling with a knock for some time.
With no further chances for either side, West Ham led at the break and the home fans vented their frustration as the teams headed towards the dressing rooms.
The toxic atmosphere inside Carrow Road soon changed as referee Jon Moss awarded Norwich a penalty eight minutes into the second half.
Jussi Jaaskelainen failed to hold a tame header from Michael Turner and, after clambering around to deal with his spillage, the experienced Finn brought down Hooper, who stepped up to tuck away the resulting spot-kick.
The lead was almost very short-lived as the Norwich defence could not deal with a ball into their box, with the returning James Collins thrashing a volley wide for the visitors.
Fer then bent an effort high over the crossbar after being afforded far too much time and space on the edge of the West Ham box as the Canaries continued to come back into the game.
West Ham enjoyed a spell on the ball before Jonny Howson went much closer in the 70th minute with a dipping long-range effort cannoning off the crossbar with Jaaskelainen well beaten.
The ball bounced out to Snodgrass, who was brought down by Collins and the Scotland winger curled the free-kick over the wall and into the net to put the home side ahead.
The goal seemed to inject new belief into the Norwich players and they continued to create chances as good work from Fer resulted in Hooper letting fly with a well-struck effort that was too close to Jaaskelainen.
The best chance to add a third was spurned by Snodgrass, who hacked at a clever knockdown by Johan Elmander when well placed in front of goal.
But the Canaries did manage to make it 3-1, with Fer tucking away a coolly taken chance in stoppage-time as Norwich sealed their third win of the season and, if Allardyce is to be believed, helped convert the pressure from Hughton on to the West Ham boss.