The Clarets sacked Sean Dyche towards the back end of the last campaign as their last throw of the dice to survive, with caretaker manager Mike Jackson unable to stop the slide into the Championship.
And 36-year-old Kompany will be Dyche’s permanent successor, after leaving R.S.C Anderlecht after three years in charge.
“Burnley Football Club is a truly historic English side and it is an honour to be appointed first-team manager,” said Kompany. “I’m excited by the challenge ahead.
“I’m looking forward to getting to work with the players and creating a positive, winning team for our fans when we return to Turf Moor.
“I’ve been impressed by the board’s vision for the club which aligns with my own and I look forward to playing my part as we enter an important season.”
Chairman Alan Pace, who was interested in taking over the Blades before his buy out at Turf Moor, added: “We are absolutely delighted to welcome Vincent to Burnley.
“Vincent is a proven leader and I’ve been very impressed with his ideas for Burnley, his appetite to succeed and his focus on leading the club back to the Premier League.
“Vincent has shown impressive credentials in leading one of Belgium’s biggest teams back to European football and a cup final last season and we’ve been excited by his philosophy, approach and ambition for the club.”