The 36-year-old joined the Black Cats in the summer from Portsmouth but has failed to force his way into manager Roy Keane's side on a regular basis and has made just eight appearances.
The former England, Manchester United and Newcastle forward will now be hoping to kick-start his trophy-laden career at Turf Moor.
"I am absolutely delighted to bring a player of Andy Cole's stature and quality to the club," said Burnley boss Owen Coyle.
"I have been working on signing him for a while now and I am thrilled to have him.
"He will be a tremendous addition for Burnley and gives us a big boost."