The Republic of Ireland international has signed a short-term deal until the end of the season and will be given the number eight shirt at the Hawthorns.
Baggies boss Roy Hodgson said he was delighted to finally sign Andrews, whom he admitted he had tried to buy when he was manager of Fulham.
Hodgson told Albion's official website: "Keith is a player I know well and have admired for several years.
"He is a flexible midfielder who can play in a number of roles across midfield. He will bring experience to the squad and also has a good goalscoring record.
"We're delighted to have him on board as we look to achieve our primary target, which is to secure a third successive season of Premier League football."
The 31-year-old has been at Ewood Park since he joined from Milton Keynes Dons in 2008, making 76 appearances for the club. He will link up with international team-mates Shane Long and Simon Cox at The Hawthorns.