Brogdon scored nine of his total in the second half and knocked down four free throws in the final 16.7 seconds to nail down Virginia's (21-5, 12-1 ACC) ninth consecutive victory.
Joe Harris led the Cavaliers with 16 points and Mike Tobey added 14 off the bench.
Clemson (15-9, 6-6) got 24 points from K.J. McDaniels, but had no other players in double figures and lost for the third consecutive try.
McDaniels nailed a 3-point jumper with 23 seconds to play that cut Virginia's lead to 59-58, but he was also whistled for a pair of subsequent fouls that brought Brogdon to the line to clinch the win.