Dawson becomes Dougie Freedman's second January signing with the Premier League side keen to send him out on loan to get more playing time.
He has been unable to dislodge Jonas Olsson and Gareth McAuley from their starting berths this season but Baggies boss Steve Clarke believes Dawson still has a big future ahead of him at The Hawthorns.
Freedman enquired about Dawson when he was in charge at Crystal Palace, and he said: "We have worked very hard on this one and he has had half a dozen teams to pick from.
"I'm very pleased he's coming to us, and not I would like to point out, for money reasons. The kid feels this is where he would like to better himself."
"I have been after Craig for a few months, to be honest. And it's paid off."