The 33-year-old moves for an undisclosed fee, having made a total of 163 Premier League appearances for Norwich City, Hull City and West Ham, contributing 26 goals and 23 assists.
“When you get the right balance in the team with left-footed players, they can really produce for you down that left side.
“He tells me he’s played holding midfield, left of a three, wide left and right, so he’s pretty versatile across midfield.
“I’ve heard from many, many people that he is exceptional in the dressing room which is a great bonus on top of his abilities.”
West Brom’s next match is a Black Country derby at Wolverhampton Wanderers on Saturday 16 January, and Snodgrass is then set for an early return to his former club three days later as the Baggies visit West Ham.