Kennedy Meeks scored 15 points and James Michael McAdoo had 11 for North Carolina (11-7, 1-4). The Tar Heels avoided the first 0-4 ACC start in team history by beating Boston College in their last game, but trailed almost throughout against the Cavaliers.
Harris did most of his scoring by connecting on a trio of 3-pointers in the first 3:14 of the game. He didn't make another shot from the field until just 2:28 remained, but he had plenty of help.
The Cavaliers used a 22-6 run spanning the halves to take a 48-33 lead. Mike Tobey, who did not score in a foul-plagued first half, had six quick points in the second half.
The Tar Heels closed within 12 several times, and then pulled to 60-49 on McAdoo's tip-in with 9:05 to play.
But Brogdon beat the shot clock with a jumper from 15 feet, McAdoo missed a pair of free throws and Akil Mitchell's driving basket pushed the lead back to 15, and it was more than enough.
Virginia made eight 3-pointers and turned 11 North Carolina turnovers into 17 points.
The first half was close until Virginia closed with a 12-2 burst to open a 38-29 lead. Brogdon made a three-point play with 1:57 left and London Perrantes, who had a career-best eight assists by halftime, swished a 3-pointer from the left corner with 27 seconds remaining.
North Carolina had nine offensive rebounds in the half, but just four points off those chances.