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.
ESPN's Dan Dakich joins The Paul Finebaum Show to discuss the SEC's presence in the NBA Draft.
Ted Kapita was the top-rated recruit from West Virginia in the class of 2015 and recently played at DME Academy in Florida.
With the Suns drafting Tyler Ulis, they now have five guards on their roster who attended Kentucky.