When the horn sounded, Belmont's players spilled off the bench to celebrate near midcourt.
Mann hit five 3-pointers and added nine rebounds to lead the Bruins, who hit 15 of 37 3s.
Windler scored 16 points and hit four 3s of his own for the Bruins, who shot 45 percent for the game and hit 20 of 22 free throws to steal back a game that looked to be slipping away late.
North Carolina ran off 14 straight points to go ahead in the second half and extended its perimeter pressure on Belmont's outside shooters, holding the Bruins to one field goal over a 10-minute span. But after the Tar Heels had pushed ahead 73-65 on Marcus Paige's driving basket with 3:46 left, Mann and the Bruins pushed back.
Mann knocked down a 3 to cut the deficit to 78-75 at the 1-minute mark, then followed with another less than 10 seconds later that pulled Belmont to within 80-78.
The Tar Heels had a chance to make it a two-possession game again, but Paige lost the ball on a dribble drive with 19.9 seconds left, leading to a jump ball that set up Mann's final shot.
James Michael McAdoo scored a career-high 27 points and had 13 rebounds to lead the Tar Heels, who again played without leading scorer P.J. Hairston and senior guard Leslie McDonald due to NCAA eligibility concerns. Paige finished with 17 points.
The Tar Heels shot 48 percent, but went just 22-for-48 (46 percent) from the foul line for the game - including a 9-for-28 performance in the first half as Belmont took a 41-34 lead at the break.