The Frenchman also added an assist in what was another good performance from the Baggies.
The striker did get on the end of Anelka's cross moments later to give Albion the lead from close range.
They doubled their lead almost immediately. Some excellent football between Dorrans and James Morrison released Anelka, who rounded the goalkeeper and fired into the empty net.
The second half saw the hosts experiment with a couple of different formations, before Shane Long had the ball in the net after some nice play with Anelka. The assistant referee's flag denied West Brom a third, though.
Both teams settled for what they had in the final 10 minutes and the game ended 2-0.