Pepperdine (13-7, 6-3 West Coast Conference) moved into sole possession of third place in conference a half-game ahead of BYU (14-7, 5-3).
Stacy Davis made two free throws and Murray hit a 3 to make it 67-61 with 1:52 left.
Kyle Collinsworth and Fischer each hit two free throws and BYU called timeout with 31 seconds left.
Raines made two free throws for a 69-65 lead before Fischer missed a 3 on the Cougars' next possession. Davis made two free throws for the game's final margin.
Collinsworth finished with 18 points, Nick Emery had 17 and Fischer added 15.