Sam Allardyce's side saw Mbaye Diagne's goal questionably ruled out by VAR early in the first half at the Hawthorns.
But they recovered from that blow to sweep into a two-goal lead by half-time thanks to Matheus Pereira's penalty and a Matt Phillips strike.
Callum Robinson sealed West Brom's second successive win after the interval as the Baggies built on their stunning 5-2 victory at Chelsea.
Second bottom Albion move within eight points of fourth bottom Newcastle with seven games left.
The Baggies have scored eight goals in their last two games, two more than they had netted in their previous 11 combined, but their unexpected revival might have come too late to save them from slipping into the Championship.