After five successive defeats, second-half goals from Robbie Brady and Wes Hoolahan put them in control against the Hammers before the visitors struck twice in the space of two minutes.
Brady broke the deadlock after 54 minutes with a curling shot that bounced over the line off the crossbar and Hoolahan doubled the lead 10 minutes later, converting from close range after Steven Naismith's shot deflected into his path.
West Ham, who could have gone above Manchester United into fifth with a victory, roused themselves and Dimitri Payet halved the deficit before Mark Noble smashed in an equaliser.
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A jittery Norwich then had to hang on desperately for a point that, thanks to Newcastle United's heavy defeat at Chelsea later, allowed them to edge out of the bottom three on goal difference.
Norwich manager Alex Neil preferred to take the positives from an improved display, rather than dwell on the last 15 minutes when it all went wrong.
"In terms of all round performance, it was certainly an improvement from where we have been. It's a style that I'm much more comfortable with," he told the Norwich website (www.canaries.co.uk).
"There was a bit of apprehension there. Everyone was so desperate for us to win the game; the players, the fans and myself."
(Reporting by Martyn Herman, editing by Pritha Sarkar)
