Steve Clarke's men ended the away side's unbeaten run thanks to the Belgium striker's late header after a dogged game of largely frustrating attacking play at the Hawthorns
In a first-half stalemate, it was the Canaries who took the lead as Robert Snodgrass saw his free kick palmed into the net by Ben Foster, with Zoltan Gera equalising for the home side just before the break.
The second half was similarly even, but the Baggies found a winner through Romelu Lukaku, who powered home a fine header in the final 10 minutes.
The Baggies welcomed Ben Foster back into their side after recovering from a groin injury, replacing Boaz Myhill in one of three changes.
Norwich made just one change from the side that beat Wigan, with Steve Morison replacing Grant Holt as the striker failed to recover from a knock.
The home side started with a great deal of intent, and saw Lukaku head wide from a superb cross by Goran Popov after two minutes.
Steve Clarke's side began to enjoy the majority of early possession, but the Canaries made it difficult for West Brom to penetrate their solid and patient defence.
It took until the 21st minute for either side to create another chance; Lukaku was played in early by James Morrison but his low shot was saved well by Mark Bunn.
The away side then gained a free kick from 25 yards out, and returning goalkeeper Foster could only succeed in palming Snodgrass’ effort into the net, giving Norwich the lead.
This seemed to galvanise the Baggies who then won a corner that was delivered well by Chris Brunt, and should have been turned home by Gareth McAuley.
The home side then created another good chance as after more fine work from Morrison; the ball was played back to the impressive Goran Popov, who blazed a fierce shot over the bar.
Chris Hughton's team were holding out well but could not manage to maintain their advantage. In the 42nd minute a Brunt corner caused havoc in their box and was eventually turned home by Gera.
It took until the 54th minute for either side to create another opportunity, as Steve Morison was played through by Alexander Tettey but saw his shot well blocked by McAuley.
From the resulting corner, a delicious cross from Snodgrass found Michael Turner at the back post, but he put his free header wide of the target.
Both sides then struggled to make any impact with their forward play, even with the Baggies having brought on Marc-Antoine Fortune and Shane Long.
A superb chance presented itself in the 71st minute as the home side won a free kick right on the edge of the Norwich area, but Peter Odemwingie blasted it into the wall much to the frustration of Brunt.
The Baggies continued to pressurise the away side, and broke the Canaries' resolve in the 82nd minute as Lukaku rose above Javier Garrido at the back post to head home and give his side all three points.