Fulham may have taken the lead but Albion were full value for the draw as the Cottagers rarely extended them.
Clint Dempsey, with his 11th goal in 13 games, opened the scoring in the second half with one of the home side’s few clear cut chances.
But Albion are no pushovers on their travels having only lost one of their last five games on the road.
So it was no surprise with eight minutes remaining that James Morrison’s flick was collected by substitute Somen Tchoyi to rescue a point.
Referee: Mark Halsey
Venue: Craven Cottage
Fulham scorers: Dempsey 69
West Brom scorers: Tchoyi 82
Yellow cards: Murphy (FUL)