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.
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point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.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.