Having dominated for much of the game the home side appeared to have grabbed a well-deserved point from the spot, yet it was stunned late on following a fine Canaries break.
A poor first half culminated in the Canaries taking the lead courtesy of a fine Andrew Surman volley against the run of play.
The second half was played at a much higher tempo as the home side leveled through Shane Long from the penalty spot, only for Steve Morison to nick all three points on the break.
Both sides made changes from their FA Cup victories a week ago. Jerome Thomas, Youssouf Mulumbu and Paul Scharner were all restored to the Baggies starting lineup, with Billy Jones, Graham Dorrans and George Thorne all missing out.
Hat trick hero Simon Cox maintained his place at the spearhead of the West Brom attack, with the returning Long starting on the bench.
For the visiting side, John Ruddy replaced Declan Rudd between the sticks.
Adam Drury, Andrew Crofts, David Fox and Grant Holt all dropped to the substitutes’ bench, with Kyle Naughton, Bradley Johnson, Anthony Pilkington and Steve Morison named in the starting XI.
A lively start saw West Brom’s Peter Odemwingie force Norwich goalkeeper Ruddy into action early on, with the shot-stopper deflecting the Nigerian’s 25-yard strike unconvincingly onto the crossbar.
His opposite number Ben Foster also had a shaky start to the game. The on loan Birmingham man rushed out to the feet of Morison and inadvertently kicked the ball against the forward, but his blushes were spared as he watched the ball roll agonizingly against the post and wide.
The home side spent much of the first half on the offensive yet was left frustrated by some stout Canaries defending.
Following a cleared West Brom corner, the ball fell kindly to James Morrison who struck a fierce half volley goalwards, only to see his effort headed off the line by the retreating Surman.
The Norwich midfielder found himself in the thick of the action again on the stroke of halftime as the Baggies were dealt a sucker punch.
Some good hold up play from Simeon Jackson left West Brom exposed at the back. Jackson laid the ball off to Wes Hoolahan who clipped an intelligent ball into Surman on the edge of the area. The former Southampton man opened up his body to drill a superb volley past the despairing Foster to give his side an undeserved halftime lead.
An uninspiring start to the second half led to Roy Hodgson turning to Long, and the Republic of Ireland international almost made an immediate impact.
After some good movement in the box, Long controlled Thomas’ cross before firing a left footed effort towards goal, only for Ruddy to save well with his legs.
The forward didn’t have to wait long to be given his second chance of the afternoon.
Norwich defender Daniel Ayala seemed favorite to reach a ball into the box before Thomas, but somehow contrived to tackle over the ball and into the winger’s leg, leaving referee Mike Dean with no choice but to point to the penalty spot.
Long stepped up to coolly send Ruddy the wrong way and restore parity for his side.
The goal lifted spirits at the Hawthorns as the home side began to search for a winner, yet the team was soon pegged back again.
Following sloppy play for Mulumbu, substitute Holt drove at the West Brom defense before reaching the byline and clipping a superb ball back to strike partner Morison, who was left with the simple task of heading his side back into the lead.
The home side pushed for an equalizer but its attempts were in vain as Norwich held on to move onto 28 points for the season.