David Nwaba added 10 points for the Mustangs (6-5), who shot 48 percent and made 9 of 12 free throws in the second half, including four in the final minute by Morgan.
Shipley had nine points in the first half as Cal Poly took a 36-32 lead.
Both teams made eight 3-pointers but Cal Poly did it on 15 attempts to 26 for Santa Clara, which shot 37 percent from the field.
The game marked the end of a five-game, 12-day road trip for the Mustangs, their longest in 35 seasons. The nation's best at protecting the ball with eight turnovers a game, Cal Poly had 13 turnovers.