Gary Hooper opened the scoring in the 13th minute and Leroy Fer sealed the contest with a well-taken finish in the final minute.
West Brom had countless chances to take at least a point from this game with Stephane Sessegnon twice failing to convert great opportunities and Shane Long also guilty of being profligate in front of goal.
It is now one win in nine games for the Baggies, and Steve Clarke's side were booed off at full-time.
The midweek hiding against a Luis Suarez-inspired Liverpool at Anfield had put Canaries boss Chris Hughton's position under further scrutiny, but he would have been delighted with the way his side responded as they leapfrogged their opponents in the table.
After starting brightly they took the lead with a wonderful strike from Hooper.
Nathan Redmond fed Fer, who picked out the striker's run with the aid of a deflection off Diego Lugano's heel. The angle of the pass was taking Hooper away from goal but he fizzed a superb strike beyond Boaz Myhill.
Albion sought a quick response with Youssouf Mulumbu, back from suspension, firing straight at John Ruddy after nice link play with Morgan Amalfitano.
The Frenchman's teasing cross from the right then saw Victor Anichebe get in a header which Ruddy gratefully grasped.
Norwich were being penned in their own half but Johan Elmander should have done better from a quick break.
The Swede's flick-on found Hooper, who quickly returned him the ball. However, Elmander then ignored his better placed strike partner as he drove forward and saw his shot blocked.
The hosts were soon back on the offensive, with Ruddy tipping a fierce Anichebe shot wide of goal, before Sessegnon contrived to fail to make any contact with a Chris Brunt corner which found him unmarked.
It summed up West Brom's first half and Clarke made two changes at the break, bringing on Long and Saido Berahino for Liam Ridgewell and, surprisingly, Amalfitano.
They helped spark Albion into a very strong start to the second half and they could have had a penalty when a Sessegnon shot, after a goalmouth scramble, struck Martin Olsson on the arm as he made the block.
They should have been level in the 52nd minute when Sessegnon surged past two Norwich defenders into the box, but the Frenchman rolled his shot wide of the post having beaten the advancing Ruddy.
At the other end Redmond's pace proved too much to handle for Uruguay defender Lugano, whose vulnerability will doubtless be noted by England manager Roy Hodgson, but the winger's pull-back could not find Hooper.
Sessegnon then spurned another chance after Olsson had gifted the ball to him six yards out, with Ruddy making the save.
Long became the latest man to miss for West Brom a few moments later.
Mulumbu's deflected shot fell to him in the area, but his shot was too tame to beat Ruddy.
Mulumbu fired an effort wide but all the home side's pressure came to nothing and they were punished as Redmond found Fer for the midfielder to cut inside and fire low beyond a flat-footed Myhill with a minute remaining.