Twine, 22, has put pen to paper on a four-year contract after the two clubs agreed an undisclosed fee.
Burnley said on their official website: “Burnley Football Club are delighted to confirm the signing of Scott Twine on a four-year contract from MK Dons for an undisclosed fee.”
Kompany said: “We are delighted to have brought Scott to the club. He’s an exciting young player that will help us in the short-term process and in the long-term project at Burnley Football Club.
“He’s a player we have looked at for a little while now, we know he can score goals and affect games. We’re glad to welcome him to the squad.”
Twine scored 20 league goals and made 13 assists last season, making 50 appearances in all competitions for the Dons after joining from Swindon for an undisclosed fee in June last year.
He added: “I’m buzzing. I have definitely made the right decision joining Burnley.
“I spoke in depth with the manager and his plans are exciting. It was a no-brainer for me to sign. I can’t wait to get started and it feels like the right place for me.”
Burnley, bidding to bounce straight back to the Premier League following relegation, appointed Kompany as Sean Dyche’s permanent replacement earlier this month.