The Tigers (15-3, 4-0 Conference USA) overcame a 15-point first-half deficit to win their ninth straight game and their 21st in a row over the Green Wave (13-6, 1-3).
Clinging to a 55-52 lead with 5 minutes left, Memphis got back-to-back 3-pointers from Chris Crawford for a 61-54 advantage with 3:58 remaining. A game-ending dunk by Johnson represented the Tigers' largest lead of the game.
Ricky Tarrant scored 16 points to lead Tulane, which led 36-30 at halftime after holding a 32-17 advantage at the 7:15 mark.
Memphis battled back to tie the game at 46. Forward Tarik Black, who sat out the first half for disciplinary reasons, ignited the charge with his defense and rebounding after entering with 16:56 to go and the Tigers trailing 41-35.
The Tigers tied the game on a three-point play by Jackson and took their first lead, 48-46, on a short jumper in the lane by Black with 11:28 to go.
Finding success in the paint, Tulane scored the game's first five points and built its lead to 15 points, using a 12-0 run to establish its 32-17 margin. Josh Davis, who finished with 13 points and 12 rebounds for the Green Wave, began the run with a layup and capped it with two free throws.
The Green Wave benefited from a strong presence on the boards, outrebounding the Tigers 17-11 in the half. Playing without Black, the Tigers allowed Tulane to grab seven offensive rebounds.
Memphis put together a 9-2 run at the end of the half to trim Tulane's lead to six, 36-30. Despite two early fouls, Tiger forward Shaq Goodwin was forced to play final 7 minutes of the half with Black on the bench.