The Republic of Ireland international has left Rovers after struggling for first-team football at Ewood Park over the last year-and-a-half and has signed a short-term contract until the summer, with the club understood to have only paid a nominal fee.
After Andrews passed his medical and selected the number eight shirt, his new manager, Roy Hodgson, spoke of his delight at signing a player he has been interested in for a while.
Speaking to the club’s official website, Hodgson said: “Keith is a player I know well and have admired for several years.
"He's 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 or our primary target, which is to secure a third successive season of Barclays Premier League football. "
The 31-year-old has 26 caps for Ireland and has an eye on playing for his country at this summer’s European Championships.