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.
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.
"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."
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.
