Terrell Vinson had 14 of his 18 points after halftime and Jesse Morgan added 13 points for the Minutemen (25-11), who advanced to next Tuesday's NIT semifinals at Madison Square Garden against the winner of Wednesday's Stanford-Nevada game.
Dartaye Ruffin scored all 17 of his points in the first half to lead Drexel (29-7), which had its 18-game home winning streak snapped. Frantz Massenat added 16 points for the Dragons, who were one of the last teams left out of the NCAA tournament.
Flint coached UMass from 1996-2001 when he took over at Drexel.
Drexel led 41-31 at halftime after Ruffin came within one of tying his career high with 18 points. And the Dragons took their biggest lead of the game early in the second half when Massenat converted a three-point play to give Drexel a 53-36 advantage with 16:26 remaining.
But the Minutemen, using full-court defensive pressure, slowly crawled back into the game. UMass finally went ahead on Morgan's 3-pointer with 7:18 left that gave the visitors a 62-60 lead and capped a dominating 9:08 stretch where the Minutemen outscored Drexel 26-7.
UMass maintained a slim lead down the stretch, reaching as much as six points on two occasions, but led 72-70 with 15.7 seconds left when Drexel called timeout with the ball.
The Minutemen could have taken a four-point lead on their previous possession, but Williams missed the front end of a one-and-one at the free throw line with 23 seconds left.
Massenat had a chance to tie it, but his 10-footer went off the back rim and Drexel couldn't get the rebound.
It will be the third trip to the NIT semifinals for UMass.