Berge had spent the last three-and-a-half seasons at Brammal Lane and helped the Blades gain promotion to the Premier League last season.
Under Vincent Kompany, Burnley have signed 11 players this summer and the fee that takes Berge to Turf Moor is £12m.
The 25-year-old made 109 appearances during his time with Sheffield United.
"I have heard a lot of good things about the club and seeing the success from outside last season, there's a great project building here," he told the club's official website.
"It's a great environment full of young talented players from all around the world, so I just think it's the perfect fit for me.
"The club has high ambitions, making several signings and it all just fits into what I'm looking for."
Kompany added: "He's an intelligent footballer, with a great eye for a pass and the ability to run with the ball."
"He also has a strength and physicality that will be invaluable in the Premier League."
Berge has won 36 caps for the Norway national team.
