Blackburn and former MK Dons manager Paul Ince had been chasing Andrews all summer. He moved for a fee thought to be £1.3M although publicly it is undisclosed.
A MK Dons statement said
"While disappointed to lose our club captain, we understand that the chance for Keith to test himself in the greatest league in the world is too good to ignore.
"We wish Keith every success in his future career and he will always be welcomed back to stadium:mk - either as a player or a friend of the club."