Nichols slipped a screen near midcourt to get free on the wing and challenged defender Tre Drye for the game-winner. He scored the last six points for Memphis (12-7, 5-3 American) over the final 2:11 and finished with 17 points and nine rebounds.
Tulane's Louis Dabney missed a heave from half court at the buzzer.
Nichols' tip-in gave Memphis a 54-51 lead with 1:43 left and Tulane (13-7, 4-4) answered 30 seconds later with Jay Hook's 3-pointer, tying it at 54.
The Tigers led 49-40 with 9:47 left, but Tulane answered with seven straight points and capped an 11-2 run tied at 51.
Markel Crawford added 15 points and five steals for Memphis.
Hook led the Green Wave with 13 points. Dabney and Jonathan Stark added 10 points apiece.