Slaven Bilic's side were frustrated in the first half, and Alex Neil's hosts took the lead through Robbie Brady nine minutes after the interval.
Wes Hoolahan added a second 11 minutes later as the canaries looked on course for a valuable win at Carrow Road, though Bilic's side had other ideas.
Dimitri Payet gave the Hammers hope 16 minutes from time as he finished after John Ruddy could only parry a shot from Victor Moses, and captain Mark Noble earned his side a point.
The skipper got onto a cutback from Payet and finished superbly from 18 yards just two minutes after Payet's goal to shock Norwich.
There were chances left in the game with Emmanuel Emenike going close at the death before Adrian brilliantly denied Nathan Redmond a 95th-minute winner at the other end.
The Hammers missed the chance to take fifth spot from Manchester United, but closed the gap on the Red Devils to just one point.
