Brandon Davies had 11 points and six rebounds for the Cougars (6-3), who rallied from a double-digit deficit in the second half.
Carlino drained consecutive 3-pointers to knot the game at 52-52 with 4:55 left. Craig Cusick gave the Cougars their first lead when he buried a 3-pointer to make it 55-53 with 4:22 remaining.
Haws sealed the comeback with a layup and free throw to give BYU a 59-58 lead with 2:48 left. He added a pair of key free throws with 16.7 seconds remaining.
Justin Seymour had a chance to force overtime with a 3-pointer, but the ball bounced off the rim as time expired.