The Baggies took a shock lead inside just two minutes when Mbaye Diagne headed home, but the Reds dominated the remainder of the half and netted a equaliser through Bruno Fernandes's sublime volley.
Referee Craig Pawson awarded the Reds a penalty just after the hour-mark for a foul on Harry Maguire before changing his mind after having another look.
Mason Greenwood and Scott McTominay went close in the second half, while Diagne had two good chances to claim victory for the hosts. Ultimately though, this was another disappointing day for United as West Brom held firm.
FIRST HALF – BRUNO PULLS US LEVEL
Ole Gunnar Solskjaer made two changes to the side that drew with Everton last weekend, with Anthony Martial and Fred coming in for Mason Greenwood, who was named on the bench, and the injured Paul Pogba.