The Cougars (20-8) never trailed after halftime but saw Utah State tie the game at 68 on Spencer Butterfield's free throws with 56 seconds to play, completing a comeback from an 11-point second-half deficit.
BYU's Brandon Davies then missed a jumper, but Tyler Haws grabbed the offensive rebound with 35 seconds left to allow the Cougars to hold the ball for the last shot. Carlino's 3-point attempt was off the mark, but Cusick cleaned up the miss and scored to help BYU pull out the win.
Haws scored 27 points for the Cougars, who won despite shooting 35 percent from the field and going 4 of 21 from 3-point range.
Spencer Butterfield had 21 points and 14 rebounds for the Aggies (18-7).
Purdue coach Matt Painter says players haven't used the fifth-year graduate transfer rule for its true intent.
The dust has settled after players made their NBA draft decisions. Which college teams won? Which lost? Well, Villanova is pretty happy. USC, meanwhile, might not be as thrilled.
The 2017 recruiting class is short on point guards, with Trevon Duval leading only eight prospects from the ESPN 60 who are still available.