The Belgian, 36, had been in charge at Anderlecht for two years before leaving by mutual consent in May, having led them to a third-place finish.
The Clarets sacked Sean Dyche in April and replaced him with interim boss Mike Jackson, who was unable to prevent relegation from the Premier League.
"I'm excited by the challenge ahead," Kompany said following his appointment.
"Burnley Football Club is a truly historic English side and it is an honour to be appointed first-team manager.
"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.