The visitors had an opportunity early on to break the deadlock as James McClean dispossessed Matthew Lowton on the left, charged into the area and fired a low-angled shot that was blocked by James Tarkowski.
On the half-hour mark, Johann Berg Gudmundsson tested Ben Foster twice, while the last five minutes of the first half saw a number of efforts from both sides as Sam Field could not readjust his position as he headed wide, while Mee missed from two yards out with the goal gaping.
The game remained goalless at the break, but just after the restart, Jake Livermore thought he had given West Brom the lead as his shot deflected off Mee into the back of the net, but it was ruled out for offside.
Nevertheless, the visitors managed to break the deadlock, Matt Phillips flicking the ball into the path of sub Robson-Kanu, who got the better of two opposition defenders before firing low past Tom Heaton.
It proved a bittersweet afternoon for the Wales forward, as he later received a straight red card for appearing to throw his arm into the face of Lowton, but the Baggies succeeded in seeing the game out and claiming all three points.