Zoltan Gera cancelled out the opener from Robert Snodgrass before Romelu Lukaku found a late winner for the Baggies to end their winless streak after four games without a victory.
Norwich began the game the brighter as the confidence from their unbeaten run was easy to see.
Steve Morison led the line with aggression and was well supported from midfield even though Albion goalkeeper Ben Foster was not tested in the early stages.
But that changed when Peter Odemwingie gave away a free-kick on the edge of his own box and Snodgrass promptly curled the ball into the top-right corner of the net.
Foster, who was returning to the side despite Boaz Myhill’s clean sheet last time out, should have done better given that he got both hands to the ball.
But the response from the home side was impressive and, with the Canaries content to play on the break, they dominated the rest of the half as Gareth McAuley saw an effort cleared off the line.
And the Baggies got their reward just before the interval when Gera poked home from close range after Norwich keeper Mark Bunn failed to clear under pressure from Odemwingie.
Chris Hughton’s men went into their shell after the break as the home side pushed hard for a winner without creating too many openings.
But Steve Clarke got his reward for introducing Marc-Antoine Fortune and Shane Long in search of three points when Lukaku powerfully headed past Bunn from Goran Popov’s deep cross.