Morgan Amalfitano scored the crucial goal after 16 minutes as Pepe Mel and West Brom ended a turbulent week on a high with a victory that leaves both sides five points clear of the bottom three.
Mel and the club's board had been forced to play down an incident that took place in the dressing room after last weekend's 3-3 draw with Cardiff City.
However, after attentions reverted to on-field matters, Amalfitano ensured West Brom responded positively on a day when fellow strugglers Cardiff were beaten by Crystal Palace.
The Marseille loanee, who put West Brom ahead against Cardiff last week, was on target once more at Carrow Road as he showed quick feet to evade the stumbling Martin Olsson before shooting across John Ruddy into the far corner.
Norwich twice hit the woodwork during the second half but there was to be no late drama, as there was when West Brom scored and conceded in stoppage time at The Hawthorns last week, with Mel's men picking up a rare away win in the top flight to keep their opponents looking over their shoulders.
Perhaps showing nerves due to the importance of the fixture, Gareth McAuley almost handed Gary Hooper a shot on goal within 30 seconds, as the former Celtic man nearly beat Ben Foster to a sloppy backpass.
From there, West Brom grew into the game and went close to an opener with strikes from Graham Dorrans and Amalfitano.
Johan Elmander then blazed an effort high and wide from inside the box, after the ball had deflected into his path, prior to Amalfitano scoring for a second week running.
Taking full advantage of Olsson's slip, the Frenchman produced a smart finish to beat Ruddy and open the scoring.
Ruddy ensured Matej Vydra did not double West Brom's lead following a quick break, while Norwich were frustrated in their attempts to find an equaliser - Foster making a smart stop to keep out a header from Hooper just after the half-hour mark.
West Brom were penned back under plenty of Norwich pressure after the interval but struggled to fashion any clear-cut chances until Hooper crashed an effort off the angle of post and crossbar from an acute angle inside the area.
Robert Snodgrass' free-kick was then tipped onto the bar by Foster but Norwich's pressure failed to bring rewards as they were beaten at home in the top flight for the first time this year.
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