Kansas became the third No. 1 seed to fall in the tournament, leaving Louisville as the only top-seeded team with a chance to reach the Final Four.
The Wolverines overcame a 14-point deficit with just under 7 minutes left in the second half with Burke pulling off the magic in the closing moments.
He hit a layup with 14 seconds remaining to get Michigan within three and then, after Elijah Johnson missed the front end of a 1-and-1 foul situation for Kansas, Burke connected on a 3-pointer with 4 seconds left that forced overtime.
The Wolverines then built a five-point lead in the final minute of extra period only to have Johnson hit a 3-pointer. Kansas' Jeff Withey followed by blocking a layup attempt by Burke with 10 seconds to go and the 24-second shot clock expired to give the Jayhawks a chance.
Johnson managed to break free for what appeared likely to be a layup that would have forced a second overtime, but he inexplicably threw the ball out to Naadir Tharpe beyond the 3-point line.
Tharpe's desperation attempt to win it from long range misfired, leaving Michigan with a chance to go to the Final Four. That chance will come on Sunday when the Wolverines (29-7) will take on Florida, which brought an end to the unlikely run of Florida Gulf Coast 62-50.
The opening semifinal game in the South Region was the first overtime game played in this year's tournament.
Mitch McGary had 25 points and 14 rebounds for Michigan while Burke scored 23. Kansas (31-6), which lost despite shooting 53 percent from the field, was led by 20 points from Ben McLemore.
All five Kansas starters scored in double figures. The Jayhawks' exit left the Big 12 Conference with no more teams in the tournament and that league wound up with a 3-5 record in the event.