Tyrece Radford added eight points and 13 rebounds for the Hokies (2-0), who didn't know they would be playing the nation's third-ranked team until early Friday morning.
The teams were tied at 29 after 20 minutes and deadlocked at 38 five minutes into the second half.
The Wildcats appeared to take control of the game from there, going on an 11-0 run and holding the Hokies scoreless for five minutes.
But Virginia Tech chipped away and took the lead at 64-62 with 1.3 seconds remaining on a traditional three-point play by Aluma, who scored on a beautiful assist from Cartier Diarra in the lane, and then unintentionally made the free throw.
Mutts made up for that mistake by scoring the first five points in extra time and the Hokies never relinquished the lead, outscoring the Wildcats 17-9 in the extra frame.
The Hokies were supposed to face Temple on Saturday, but the Owls were forced to pause team activities for two weeks on Thanksgiving after a positive coronavirus test in the program. That meant canceling not only Saturday's game, but a contest that had been scheduled for Dec. 3 at Villanova.
That left both the Hokies and the Wildcats at the Mohegan Sun looking for a game.
Virginia Tech: Neither team has anyone over 6-foot-9 on the roster, but the Hokies outscored Villanova 32-22 in the paint and outrebounded the Wildcats 40-31.
Villanova: The Wildcats' bench scored just eight points against the Hokies, who got 25 points from their reserves.
Virginia Tech: The Hokies stay in "Bubbleville" for a Sunday night game against South Florida.
Villanova: The Wildcats head home to play Saint Joseph's, a game that was moved from Monday to Wednesday as part of the schedule reshuffling resulting from Temple's virus issues.