The most recent encounter between the Bluebirds and the Baggies was in March 2014, a 3-3 draw at The Hawthorns.
Albion went two goals ahead in the opening ten minutes as Morgan Amalfitano and Graham Dorrans found the net.
Jordan Mutch scored to pull one back for the visitors, with Steven Caulker then levelling with less than 20 minutes to play.
Thievy Bifouma netted what looked like a winner for Albion in stoppage time, but later still Mats Daehli once again restored parity.