The Waves (10-5, 3-1 West Coast Conference) never trailed and made all the clutch plays after the Cougars (13-5, 3-2) cut a double-figure deficit to one with 2:03 left.
BYU had a chance to take the lead but Tyler Haws, who led BYU with 26 points, missed a jumper at 1:06.
After Davis made two free throws at 41 seconds, the Cougars, who were 1-12 from 3-point range in the second half, missed. Jeremy Major made two free throws at 23 seconds and then got a breakaway layup for a 66-59 lead.
BYU came in averaging 88.3 points while Pepperdine allowed 62.
Kyle Collinsworth, who had his third triple-double in BYU's last game, had four points and three rebounds.