Dragan Kesich, whose 30-yarder with 3:40 to go gave the Gophers the lead, missed his chance to win it as time expired when his 47-yard try went wide right. He missed a 27-yarder earlier.
Preseason All-American Omarion Hampton rushed 30 times for 129 yards and Jahvaree Ritzie had three sacks for the Tar Heels, who overcame a rough first half and a disheartening injury to their new quarterback in the first game of the post-Drake Maye era.
Minnesota newcomer Max Brosmer had a rushing touchdown and went 13-for-21 passing for 166 yards, but his lost fumble on a first-down run early in the fourth quarter set up Burnette's first of two go-ahead kicks.
With Darius Taylor sitting out with a leg injury, Marcus Major, who transferred from Oklahoma, rushed for 73 yards and a touchdown in his Gophers debut.
Justin Walley had a 70-yard interception return that set up Major's score to give the Gophers a 14-7 halftime lead the Gophers had a golden opportunity to build much bigger.
Za'Quan Bryan dropped an interception at the North Carolina 16 when Johnson's intended receiver J.J. Jones fell down and the ball went straight into the cornerback's arms. Kesich's short kick rattled off the right upright. Then Johnson's fumble at his own 10 was recovered by Minnesota, until a holding penalty on Aidan Gousby negated the turnover. The Tar Heels went on an 80-yard touchdown drive.
North Carolina hosts Charlotte on Sept. 7. The Tar Heels have never played the 49ers, who are in their 10th season in the FBS.
The Associated Press contributed.