Carter hit a 3-pointer midway through the second half to give the Irish (19-6, 7-5 Big East) a 62-57 lead and spark the run. The Irish moved ahead 70-59 when Kurz made a pair of free throws with 3:42 left.
Providence (15-9, 5-6), which managed just five points from 12:18 left in the second half until Brian McKenzie hit a 3-pointer with 3:26 left, trailed 73-62 when Falls made a pair of free throws with 1:36 remaining. But the Friars used a 16-8 run down the stretch to make it close.
Providence closed to 78-75 when Weyinmi Efejuku made a 3 with 10 seconds left. Carter made a free throw with seven seconds left and the Irish put the game away when Zach Hillesland stole the ball at midcourt with five seconds left.
The Irish improved to 16-0 at home this season, while the Friars fell to 1-6 on the road.
Hillesland added 14 points for the Irish, who shot 55.2 percent. Herbert Hill led Providence with 18 points and Jonathan Kale had 17 points, but just two points in the second half. McKenzie added 16 points and Sharaud Curry had 11.
The Friars shot 52.5 percent and outscored the Irish inside 46-38. But after turning the ball over just five times in the first half, the Friars had 13 turnovers in the second half.
Providence led throughout the first half, jumping to a 7-0 start. The Friars made nine of their first 14 shots, moving ahead 24-12 when Herbert Hill capped a 9-2 run with a basket inside midway through the half. The Irish closed to 40-38 with 1:23 left in the half when Falls drove inside for a basket.
The Irish had a chance to go ahead, but Falls missed a 3, and he was called for a blocking foul with 28 seconds left that the fans disagreed, booing loudly until the officials left the floor for intermission and booed again when they returned for the second half. Sharaud Curry made both free throws to give Providence a 42-38 halftime lead.
Notre Dame became the 11th Division I program with 1,600 all-time victories.