Mann drained a trio of 3-pointers in the final 3 minutes, helping the Bruins (3-1) erase an eight-point deficit and post an 81-80 lead.
The Tar Heels' J.P. Tokoto missed a shot in the closing seconds and Belmont's Caleb Chowbay then put away the win with a layup.
Drew Windler added 16 points as the Bruins logged their first-ever victory against an ACC team.
North Carolina (2-1) had its streak of 82 straight home wins against unranked non-conference opponents snapped with the stunning loss.
James Michael McAdoo provided 27 points with 13 rebounds and Marcus Paige added 17 points and five assists for the Tar Heels.