Grovey led TU with 16 points and Mayberry scored 11.
Tiffani Couisnard had a big night on the boards.
The Cougars led 28-26 at halftime, but Tulsa came back and took the lead for good when Grovey hit a 3 with 6:14 left in the game.
Another 3 by Kadan Brady gave a TU a seven-point advantage at 56-49 with 3:17 left, matching the largest lead it had attained earlier in the game.
Tulsa shot 39 percent (20-of-52) from the field, 47 percent (7-of-15) from 3-point range and 71 percent (12-of-17) from the free throw line.
Houston finished with 38 percent (21-of-55) from the floor, shot 70 percent (7-of-23) in 3s and connected on 46 percent (6-of-13) of its free throws.
TU finished with a 42-33 rebounding edge over the Cougars.