The Hammers skipper scored either side of the break as his side moved into fifth place in the Premier League table, although Manchester United could regain that spot on Sunday with a positive result against Leicester City.
The first opportunity of the game came courtesy of the trickery of Jonathan Leko, who was making his first start for the Baggies, but the 17-year-old's cross was not converted by Craig Gardner, who saw two shots well blocked.
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Just before the quarter-hour mark, Gardner was denied by Adrian who tipped a well-struck effort wide of the post before West Ham finally created an opening at the other end with Andy Carroll heading harmlessly over the crossbar.
Leko and Gardner again combined with the older of the pair heading wide of the target, but West Ham improved as the half progressed and with 11 minutes before the break, they opened the scoring when Cheikhou Kouyate was able to head home from a cross from Dimitri Payet.
West Brom found themselves two goals adrift in added-on time of the first half when Kouyate turned provider to cross for midfield partner Noble, who controlled the ball before prodding it past Ben Foster in the Baggies goal.
Chances were few and far between after the restart but it was West Ham who remained in the ascendancy, with Carroll failing to find a teammate with a header inside the penalty area after being set up by Aaron Cresswell.
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West Brom improved and had an opportunity through substitute Saido Berahino which was sent wide but the Baggies failed to build momentum and with just over 10 minutes left, the Hammers sealed the three points.
Payet linked up with Carroll who delivered a teasing cross for Noble, and the midfielder was ruthless in firing the ball into the roof of the net on the volley.
The final 10 minutes were played out at a pedestrian pace but it mattered little to the visitors who remain on course to claim a spot in next season's Europa League.