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.
Attkisson leaves CBS News, reportedly over network's 'liberal bias'
McPhee, Cokas 'working on their marriage' after affair