The results left Fulham and Blackburn in the bottom three after four matches, although Blackburn at least picked up their first point after three successive defeats.
Blackburn stayed bottom, one place behind Swansea City, who like Fulham have two points. West Brom, who started the day in 19th spot, leapt to 12th.
Blackburn took the lead at Craven Cottage with a stunning left-foot shot from Spaniard Ruben Rochina after 32 minutes and Bobby Zamora equalised, also with a left-foot shot, six minutes later.
The visitors finished with 10 men after Canadian youngster David "Junior" Hoillet was carried off on a stretcher after a stoppage-time collision with Fulham goalkeeper Mark Schwarzer.
"He is up on his feet and conscious but he was out for a few minutes and needs to get stitched up," Blackburn manager Steve Kean told Sky Sports.
Kean has incurred the wrath of Blackburn fans who have written an open letter for him to be sacked.
"That doesn't affect me," he said. "I've got the backing of the players and the owners and if the fans stick with us I am sure they will see many away performances like that which will give us more than one point."
West Brom picked up their first win of the season after Peter Odemwingie scored after only three minutes at Carrow Road.
The 30-year-old Nigeria international striker, back in the side after injury, took advantage of a mixup between central defender Richie de Laet and reserve goalkeeper Declan Rudd to score.
Odemwingie's second-half penalty was saved by Rudd and Norwich, back in the top flight after six years, have given away a penalty in each of their four matches this season.
All the main title contenders played on Saturday when Manchester United crushed Bolton Wanderers 5-0 and Manchester City beat Wigan Athletic 3-0 to maintain their perfect starts to the season.
Queens Park Rangers host Newcastle United on Monday.
