The Express & Star exclusively revealed earlier this week that the midfielder was closing in on a move to The Hawthorns.
Snodgrass underwent his medical yesterday and has agreed terms on an 18-month deal.
“I am absolutely delighted he has chosen to take up the challenge with us and that we will be adding some much needed Premier League experience and quality into the squad," said manager Sam Allardyce said.
“I’m delighted he has said yes. He is left-footed. We all look for balance.
“I think it’s good to have left-footed players who can produce on that side rather than right-footed players who have to check inside, that can sometimes lose you an opportunity.
“And he’s got experience. He tells me recently he has played in central midfield in a left of a three as well as wide left and wide right.
“He is flexible in terms of the number of the positions he can play which I think is very important.
"And I’ve heard from many people he is exceptionally good in the dressing room which is a great bonus on top of his ability.”