The Wolverines (23-4, 10-4 Big Ten) shook off a sluggish first half with a strong second half to win consecutive games for the first time in nearly a month.
"I know we're back on track," Burke said.
The Fighting Illini (20-9, 7-8) had won five straight -- a streak that started Feb. 7 by beating top-ranked Indiana -- and looked like they might earn another impressive win when they led by eight points twice in the first half.
In the second half, though, Illinois coach John Groce said Michigan was tougher and smarter.
"It's unacceptable," Groce said. "It's intolerable."
Michigan opened the second half with a 15-3 run, going ahead for good on Burke's 3-pointer with 16:49 left to play and had a nine-point lead with his two free throws that put him over the 1,000-point mark for his career.
The Illini pulled within four points with 7:55 left, but Burke dashed their comeback hopes.
Burke assisted Tim Hardaway Jr. on a fast-break layup, made a 3-pointer and pumped his fist and screamed with joy when his pass led to Caris LeVert making a shot beyond the arc in a spectacular, 63-second stretch.
Groce was impressed, but not surprised because he has been watching the native of Columbus, Ohio, play since he was in middle school.
"The thing that he's always had is competitive toughness," said Groce, who led the Ohio Bobcats past Michigan in last year's NCAA tournament. "He's had that since Day 1."
Burke joined Phil Hubbard, Mike McGee, Jalen Rose, Chris Webber, Louis Bullock and Manny Harris as the sophomores who have scored 1,000 points at Michigan. Cazzie Russell, Bill Buntin, Rudy Tomjanovich and Henry Wilmore reached the milestone in their second seasons after they were ineligible to play as freshmen.
"It's definitely an honor to be in that company," Burke said.
Wolverines coach John Beilein said the feat is even more impressive because Hardaway Jr. is in the 1,000-points club, too.
"It's not like he's a one-man show," Beilein said.
Michigan played for the first time in a week and looked rusty and sloppy early, turning the ball over the missing open shots.
Illinois led by as much as eight points in the first half and didn't trail until Hardaway Jr. had a three-point play with 3:30 to go. Brandon Paul scored all 10 of points in the first half to help the Illini lead 31-28 at halftime. Paul didn't attempt a shot in the second half. D.J. Richardson and Tracy Abrams each scored 10 points for the Illini.
Hardaway Jr. scored 13 and Glen Robinson III had 10 points. Freshman Nik Stauskas, who was averaging 12-plus points, missed all five of his shots and was held scoreless for the second time this season. LeVert, another freshman guard, took advantage of his opportunity to play more, scoring eight points, one shy of his career high.
"He was a buzz off the bench and helped us open the lead up," Burke said.
Tyler Griffey's second 3-pointer pulled Illinois within four points with 7:55 left in its final push to win the game.
"Coach talked to us at halftime about how we came out strong in the second half and beat Indiana, and how we came out flat and lost to Michigan State," Griffey recalled. "We all listened, and then we came out and played like that in the second half. They really amped up the intensity and we didn't respond at all."
Burke helped the Wolverines close the game strong after starting they looked lackluster following their layoff.
"Thrilled with the second half," Beilein said. "That second half, we looked a lot like we did earlier this year."
Michigan plays Wednesday night at Penn State before hosting rival Michigan State next Sunday in a rematch after losing to the Spartans by 23 points earlier this month.