Kendall Timmons added 12 points to help Tulane (13-5, 1-2) snap a five-game losing streak against the Hurricane.
Trailing 54-49 with 9:37 left to play, Tulane went on an 18-7 run to take a 67-61 lead with 3:44 remaining. Tarrant sparked the run with a pair of free throws, then capped it with consecutive 3-pointers.
Pat Swilling Jr. made a 3-pointer for Tulsa (11-8, 3-2) with 39 seconds left to cut the deficit to one, 71-70.
After Davis made a pair of free throws with 16 seconds to go, making it 75-72, James Woodard missed a game-tying 3 for Tulsa with 4 seconds left.
Woodard led the Hurricane with 21 points and seven rebounds.