The Blue Devils (5-3 Atlantic Coast Conference, 16-5) beat the Tar Heels (3-5, 12-6) for the first time since 1993 in a game that featured 17 lead changes and seven ties.
Neither team led by more than seven points in the 197th meeting of the storied rivalry.
Landgon's previous high was 25 points this season against Clemson. Jeff Capel was also big down the stretch for Duke, scoring 19 points and making several big defensive plays.
Serge Zwikker led the Tar Heels with 14 points, while ACC scoring leader Antawn Jamison was held to a season-low 10 points.
Duke fell behind 54-48 with 12:19 left as floor leader Steve Wojciechowski picked up his fourth foul and went to the bench. But the Blue Devils rallied with their point guard sidelined for more than four minutes, regaining the lead at 56-54.
Near the end of Duke's run, Jamison picked up two offensive fouls 18 seconds apart and also had to go to the bench for a brief time with four fouls, setting the stage for a frenzied final six minutes.
The lead changed hands five times on successive possessions before Capel's driving layup with 3:54 left gave Duke the lead for good at 67-66. Less than a half a minute later, Capel blocked a 3-point attempt by Ademola Okulaja at the top of the key and received an outlet pass from Greg Newton for another layup as Duke went up by three.
But North Carolina was not dead yet. The Tar Heels closed to 72-70 on a 12-foot jumper by freshman Ed Cota with 1:33 remaining. After a jump ball possession that went to Duke and a timeout, Langdon nailed his career-best seventh 3-pointer, and the Blue Devils sealed it in the final seconds from the foul line.
The Tar Heels were rattled early, turning the ball over on their first five possessions as Duke grabbed a seven-point lead with its small lineup.
Newton, Duke's leading scorer who had started all 20 games this season and 51 in his career, didn't start the game, which began as a chess match of Duke's quickness against North Carolina's tall front line.
After a quick start, Duke hit a cold spell against the zone, going 5:56 without a point.
Five different North Carolina players scored during a 13-3 run as the Tar Heels grabbed a 24-21 lead 8:18 before halftime.
The Blue Devils missed eight shots and turned the ball over five times during their scoring drought.
From that point the score was tied four more times before Duke took a 33-31 halftime lead.
Both teams had trouble holding onto the ball in the opening 20 minutes as there were a combined 24 turnovers committed by 14 different players.