Jay Rodriguez struck after 36 minutes to push Sean Dyche's into ninth place and keep their slim European hopes alive.
David Moyes' side had 22 shots throughout the match, five of which were on target.
West Ham remain four points above the drop zone, although Bournemouth and Aston Villa have a game in hand.
They will travel to Carrow Road on Saturday in a match that could see Norwich relegated to the Championship.
Speaking after the game, Moyes said: "Frustration is probably the word I would use more than anything, the missed opportunities. I am disappointed that we didn't take something from the game because I feel we warranted something.
"I thought today was a must-win, I thought Newcastle was too, Chelsea we had to do the best we can and we did. We have been in that situation for a while. In fact, probably since I took over."