It was the Longhorns' ninth win in the last 10 meetings against the Tar Heels.
Coleman finished with 22 points and was the tournament's most valuable player for the Longhorns (4-0), who blew a 16-point lead late in the first half and fell behind with about 2 minutes left. Coleman and Kai Jones both came up big in the critical moments for Texas, securing the Longhorns' first Maui title in their fifth appearance in the tournament.
Jones had 12 points, including a tying jumper with 2:19 left and then a transition dunk with 41.2 seconds remaining that pushed Texas to a 67-65 lead. UNC's Leaky Black answered with two free throws with 25.6 seconds left, only to see Coleman make a tough shot against RJ Davis that hit the rim, bounced off the backboard and dropped through the net.
Garrison Brooks had 18 points for the Tar Heels (3-1) despite playing through an ankle injury that sent him to the locker room at least twice.
The Longhorns - who had locked down on Indiana a day earlier - asserted control midway through the first half. Texas used a 26-5 run to turn a five-point deficit into a 36-20 lead on Greg Brown's free throws with 3:50 left.
But the Tar Heels roared out of halftime with five straight scoring possessions, the start of an effort that saw them do a better job of getting the ball inside and avoiding turnovers. They clawed to within one five different times before finally tying it on Caleb Love's free throw with 3:54 left, and then took their first post-halftime lead on Brooks' contested turnaround with 2:35 to play.
Jones answered a tying jumper of his own on the next play, setting up the deciding final minute.
The tournament was played in the North Carolina mountains instead of its traditional Hawaii setting due to the coronavirus pandemic. That gave the Tar Heels three games in coach Roy Williams' hometown, though with fan cutouts in the stands and crowd noise pumped in.
The Longhorns get another test Sunday when they host No. 12 Villanova in the Big 12-Big East battle.
The Tar Heels look to bounc back against No. 3 Iowa on Tuesday in the ACC/Big Ten Challenge.