Asumnu, who scored just four points in the game, rebounded a C.J. Watson shot from the baseline on the Vols' final possession following a timeout by the UT bench.
Jon Higgins, who scored 14, fueled the rally with two big baskets to tie the game at 62-62 (1:31) and at 64-64 at the 1:00 mark. UT's defense forced an Ole Miss shot clock violation with 23 seconds to go before Asumnu's winning basket.
Ron Slay, who fouled out with 2:11 remaining and the Vols trailing 62-57, scored 17 points. Watson scored a career-best 17.
Aaron Harper led Ole Miss with 23 points.
Tennessee trailed by as many as 12 early in the first half before rallying to take a 30-29 lead at halftime at Tad Smith Coliseum.
The hot-shooting Rebels, who hit 7 of their first 10 field goal attempts, took a 23-11 advantage on a David Sanders layup with 9:52 remaining in the half.
Tennessee took its first lead at 28-27 on a John Winchester three-pointer with 2:41 to go before halftime to cap a 17-4 UT run over a seven-minute stretch.
The Vols will travel to Gainesville on Wednesday to play the high-powered Florida Gators. The game is scheduled to began at 7pm.