McRae was held scoreless until the final three seconds of the first half, but he heated up in the second half and broke a 50-50 tie with just under a minute left. The Volunteers (10-8, 2-4 Southeastern Conference) led 54-50 before Trevor Releford hit a 3-pointer with 18.5 seconds remaining.
Alabama (12-7, 4-2) had one last chance to win the game after Trae Golden missed the front end of a 1-and-1 opportunity with 15.4 seconds left, but the Crimson Tide failed to capitalize.
On their final possession, the Tide worked the ball to Trevor Lacey, who left his feet and appeared to draw contact from Tennessee forward Jarnell Stokes. But no foul was called, allowing the Vols to clinch the win.
Stokes had 15 points and a career-high 18 rebounds for Tennessee. Releford scored 18, Lacey had 11 and Rodney Cooper added 10 for Alabama.
McRae entered the day averaging 21.6 points per game in SEC competition, ranking second in the league. The 6-foot-5 junior had scored 21 points in a 68-65 loss at Alabama two weeks ago. But he missed his first five shots Saturday.
Alabama seemed intent on stopping McRae at all costs and daring anyone else in Tennessee's lineup to pick up the slack. No one seemed ready for that challenge early on, as the Vols shot 32.1 percent (9-of-28) in the first half and missed all five of their 3-point attempts.
Tennessee didn't get called for a foul in the first 10½ minutes of the game and gave up only one offensive rebound in the first half, but the Vols still trailed 25-19 at halftime because of their shooting woes.
McRae's basket late in the first half finally gave them some momentum. That shot was part of a 9-0 run that carried over into the second half and helped Tennessee erase a 10-point deficit.
McRae sank a 3-pointer on Tennessee's first possession of the second half. The Vols eventually tied the game 26-26 with 17:56 left when Kenny Hall converted an offensive rebound off McRae's missed 3-point attempt.
Alabama answered with six straight points to move back in front. The Tide stayed ahead for most of the second half and led by as many as seven.
Tennessee finally broke through down the stretch. Stokes' basket with 3:31 left gave UT its first lead since the game's opening minute. After the Vols stopped Alabama on its next possession, Stokes capped an 8-0 run by hitting a turnaround jumper that bounced off the right side of the rim and fell through as the shot clock expired with 2:28 left.
Just when it appeared Tennessee had seized control for good, Lacey sank a 3-pointer that tied the game at 50 with 1:58 left.
McRae missed a 3-pointer on Tennessee's next possession, but Stokes grabbed the offensive rebound and the Vols called a 30-second timeout. They worked the ball back to McRae, who drove into the lane and scored with just under a minute left.
Golden hit both ends of a 1-and-1 to extend the lead to 54-50 with seconds left, but Releford answered with his 3-pointer.
That set up the frantic finish that enabled the Vols to prevail.