The Scarlet Knights (14-15, 7-11) went ahead with 12.2 seconds left when Tyson Acuff made two free throws after being fouled on a 3-pointer, but they couldn't make one last stop.
Rutgers, which led by 13 in a 106-point first half, was ahead 74-62 midway through the second.
Michigan went on a 14-2 run to tie it with 4:33 left and made enough shots and stops to win.
Vladislav Goldin had 22 points and 11 rebounds, Burnett finished with 20 points, Will Tschetter scored 16 and Danny Wolf added 10 points and eight rebounds for the Wolverines.
Rutgers freshman Lathan Sommerville scored 10 of his season-high 17 points in the second half when the highly touted players in his class, Dylan Harper and Ace Bailey, cooled off. Harper finished with 17 points and Bailey had 13 after both scored 11 in the first half, helping their team score 57 points in 20 minutes.
Takeaways
Rutgers: Steve Pikiell has his highest-scoring team in nine seasons at the school, but the Scarlet Knights are not playing the type of defense they usually have under their coach.
Michigan: The Wolverines made just five shots beyond the arc for the third straight game, attempting 20-plus each time.
Key moment
Burnett's 3-pointer from deep on the left wing won the game just before time expired.
Key stat
Rutgers shot 68% in the first half.
Up next
Michigan hosts Illinois on Sunday, and Rutgers visits No. 20 Purdue on Tuesday night.
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Credit: AP
Credit: AP
Credit: AP
Credit: AP
Credit: AP
Credit: AP
Credit: AP
Credit: AP