Slay finished with a career-high 31 points for the Volunteers, who trailed by as many as 17 in the first half. Higgins had a career-high 20 points and Brandon Crump added 15.
The Bulldogs led by three points with 2:51 to go after Tim Behrendorff made two baskets.
But Crump scored inside to cut the lead to one with 2:40 left, setting up Higgins' heroics. Slay's free throws sealed it.
The Bulldogs led by as many as 17 in the first half and were ahead by 11 early in the second until the Vols made an 18-8 run to take a one-point lead.
Slay went around Simon Conn for a basket and Higgins followed with a 3 to start the run. Another 3 by Higgins tied it at 51, but Tennessee didn't take its first lead since the opening minutes until Higgins got a steal and scored to make it 58-57 with 8:42 left.
Gardner-Webb took back the lead and went ahead by 3 twice in the next five minutes until Higgins hit another 3 with 3:48 left to put Tennessee back up 64-63.
Slay played for the first time since Jan. 19, when he tore a knee ligament and missed the last 15 games of last season.
Otis Daniels led the Bulldogs with 16 points while Behrendorff added 15, Eli Strait had 13 and Josh Chiles 12.
Gardner-Webb, only in its third year of Division I-A membership, used accurate three-point shooting and aggressive rebounding to take an early lead.
A 16-3 run capped by 3s from Strait and Brian Bender put Gardner-Webb ahead 21-8 with 11:51 left before halftime.
Gardner-Webb's zone defense and big post players forced the Vols into 13 three-point attempts. The Vols made just one and shot 10-of-33 from the floor in the first half.