The 35-year-old Slovakia international moves to Turf Moor on an initial month's loan, having failed to feature for the Black Cats this season.
Clarets boss Owen Coyle told the club's official website: "I'm delighted to get Stan in on loan.
"I have known him for years, both from when I played against him and during my time as a coach and manager.
"In light of our injuries and suspensions, he will bring a physical presence to the backline. I'm sure he will complement the squad and have a big impact on the team."
Varga is in his second spell with Sunderland, having originally joined the club in the summer of 2000. He has also played for Celtic and West Bromwich Albion.