The Black Cats striker, who spent two months on loan with The Clarets at the end of last season, is moving to Turf Moor for an initial £750,000 fee although that will increase to £800,000 depending on appearances.
"We were fortunate enough to get him on loan first and he was like a new signing to us last season," said Burnley boss and former Sunderland coach Steve Cotterill.
"Having him at Turf Moor for those two months helped him to bed in ahead of next season and we are delighted."