The Toreros (1-3), who gave up 112 points in their first three games, dominated the Bulldogs (2-2) to prevent their worst start since 1959, when they began 0-4.
Butler was held to just 91 yards and eight first downs. Toreros defensive end Mario Kurn, who came into the game with seven sacks, added three more.
Morelli scored on runs of 1 and 3 yards, and Mason Mills hit Reggie Bell with a 6-yard TD pass. Mills finished 18 of 22 for 198 yards with one interception.
San Diego has won eight of its last nine games against Butler, and is 9-1 at home all-time against the Bulldogs.