The centre-back, 21, initially moved to Turf Moor on loan until January during September's transfer deadline day.
Keane has impressed during his time with Sean Dyche's men, making 10 Premier League appearances, eight of which were starts.
The England Under-21 international has now penned a three-and-a-half year deal at the Lancashire club.
Speaking about the signing, which marks Burnley's first foray into the market this month, Dyche said: "Michael is a good young player.
"We see a lot of development in him as he's on a good pathway at the moment and we're delighted that he has decided to join us."
Having started his career at United, Keane struggled to break into the first team at Old Trafford and made just five senior appearances for the club.
Subsequently, he has enjoyed loan spells with Leicester City, Derby County and Blackburn Rovers before his move to Burnley.
After initially acting as back-up, Keane grasped his chance when veteran defender Michael Duff sustained an injury in November.
