Burnley secured promotion last month when the club returned to the top flight after one season, sealing the Championship title with two games to spare.
"Burnley and Turf Moor felt right pretty much from the start, so it also feels right to sign for the next five years," former Manchester City and Belgium defender Kompany said in a statement.
"Together with the fans we have made Turf Moor a fortress again and continue to look forward to the future and the job of making Burnley better with every step."
Burnley will wrap up their season on Monday with a game against Cardiff City at home, where they will be presented with the trophy.
"We are ecstatic Vincent has signed a new deal. He is the perfect fit for how we all see Burnley Football Club moving forward," Burnley chairman Alan Pace said.
"An extraordinary leader, setting ever higher standards and driving our club to the levels we all want to achieve."
Burnley's success has also seen new investment come in from former NFL player JJ Watt (34) and his wife Kealia Watt (31), a former US international football player.
