Leading by one with 12 seconds remaining, Matt Carlino hit a clutch three as the shot clock expired to seal a 63-61 win for BYU.
BYU almost let the game slip away after Andon Winder fouled Jordan Baker as he was shooting a three, giving him three free throws.
The Cougars trailed by 11 at the half, but Brandon Davies notched a double-double, scoring 20 points and 12 rebounds, and Tyler Haws scored 15 second-half points to finish with 21 on the night.
BYU struggled in the first half, going 0 for 6 from beyond the arc, but after some second half adjustments, the Cougars netted 6 three pointers.
Davies controlled the game for BYU but got in some foul trouble early in the second half. Carlino took over for the Cougars, including a pair of clutch free throws to keep BYU in the game.
Pepperdine's Nikolas Skouen finished with 16 points for the Waves.