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.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.