Illinois, Nebraska and Iowa also picked up first-round victories in a tournament that has five nationally ranked teams participating.
Despite its high national stature, Michigan is seeded only fifth in this event and the Wolverines appeared in trouble at halftime when they held only a two-point lead.
Burke, however, finished with 21 points and both Tim Hardaway and Nik Stauskas had 15 in pushing Michigan (26-6) into the quarterfinals for a meeting with No. 23 Wisconsin.
Michigan used a 25-7 run in the second half to put the game away.
Illinois recorded a thrilling 51-49 victory over Minnesota in the opening game of the day and Nebraska pulled off a mild surprise with a 57-55 decision over Purdue.
Brandon Paul hit a jumper at the buzzer to give Illinois the win. The Illini (22-11) will next take on No. 3 Indiana.
Illinois (22-11) tied its game with Minnesota on a 3-pointer by D.J. Richardson with 48 seconds to play, a basket that came after Illinois had grabbed four straight offensive rebounds.
Paul scored 25 points to lead Illinois while Austin Hollins had 16 for Minnesota. The Gophers (20-12) lost their third in a row and dropped their seventh in 10 games.
Nebraska (15-17) owned a six-point lead over Purdue with 1:30 to play and held on to set up a quarterfinal showdown with No. 9 Ohio State.
The Cornhuskers moved in front by two when Ray Gallegos made 1-of-2 from the line with 13 seconds left and Purdue's Terone Johnson then misfired on two chances to tie the game.
Iowa earned its quarterfinal date with No. 7 Michigan State by powering past Northwestern 73-59.
The Hawkeyes (21-11) built a 14-point halftime lead and picked up the victory despite making only 37 percent of their shots from the field. Roy Devyn Marble led Iowa with 19 points while making only 4-of-12.