Cheikhou Kouyate headed in Dimitri Payet's cross after 34 minutes and skipper Noble doubled the visitors' lead in first-half stoppage time from Kouyate's pass.
West Brom offered little in response and Noble added his side's third late on with a volley as West Ham stretched their unbeaten run to 10 league games, moving above Manchester United on goal difference into fifth place with 59 points.
The London club's previous best points haul in the Premier League era was in 1998-99 when they came fifth with 57 points.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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