Tournament MVP Joe Harris added 15 points and hit the backbreaking 3-pointer with just under 2 minutes remaining for the top-seeded Cavaliers (28-6).
They shot 45 percent, pestered the Blue Devils into 38 percent shooting and used a late 12-3 run to pull away and claim the second ACC tournament title in school history.
Jabari Parker scored 23 points on 9-of-24 shooting for the third-seeded Blue Devils (26-8), who were making their 31st appearance in the championship game but were denied their ACC-record 20th title.
Anthony Gill added 12 points and was 10 of 17 from the free-throw line for Virginia. Duke as a team was just 7 of 11 from the line.
The Cavaliers came to Greensboro off of their first outright regular-season title since 1981, and were trying to do something not even Ralph Sampson could do.
It had been a long time -- 38 years -- since Wally Walker led Virginia to its only previous ACC tournament title.
Now that trophy will have some company.
With its "pack line" defense clicking and holding Duke 13 points below its scoring average, Virginia was in control for most of the day, yet could never get any separation until the final 2½ minutes.
Harris gave Duke a final chance when he rushed up a shot and Quinn Cook countered with a 3 with 1½ minutes left to pull the Blue Devils back within four.
Brogdon then drove through the Duke defense for a momentum-shifting layup that made it 66-60 with just over a minute to play and caused Duke assistant Jeff Capel to tellingly slump back on the bench in exasperation.
Cook missed a short jumper before Brogdon hit six of eight free throws in the final minute to help the Cavaliers pull away.
Hood finished with 13 points on 4-of-12 shooting and Amile Jefferson added 11 for Duke.
This was a one-possession game throughout the second half until Mitchell's tip-in with 5:10 left pushed Virginia's lead to 57-53.
Parker missed three shots on Duke's next possession, then picked up his fourth foul with 3:53 left and Gill hit two free throws to put the Cavaliers up by six.
Virginia was one of the few opponents which found a way to slow Parker, holding him to eight points on 3-of-11 shooting in the regular-season meeting -- a four-point Duke win that came down to the final seconds.
With Mitchell keying the defensive effort on him, Parker was just 2 of 10 in the first half.
Then he got hot midway through the second half and single-handedly kept Duke in the game, reeling off seven straight points for his team in a 90-second span.
He went coast-to-coast and threw down a one-handed dunk on London Perrantes, hit a 3-pointer from the wing to briefly give Duke a one-point lead and added a jumper off a spin move with 8½ minutes remaining.
Mitchell finished with 15 rebounds for Virginia, which led 28-25 at the end of a physical, intense first half notable for its low shooting percentages -- and a technical foul on Duke coach Mike Krzyzewski.
Official Jamie Luckie hit Coach K with a T with 2:07 left in the half when Krzyzewski, with his back to the court, tossed his dry-erase marker toward his bench while coming out of a timeout.
There were plenty of reasons for Duke to be frustrated, yet the Blue Devils managed to keep themselves in the game despite a rough first half from its two best players: Parker and Hood were a combined 4 of 18 from the field.
Parker hit a point-blank shot while falling down with just under 16 minutes left to put Duke up 34-33 -- its first lead since the opening minute.
Brogdon snatched the lead right back with a jumper on Virginia's next trip down the court, and it stayed tight the rest of the way until the Cavaliers' late surge.