Jerrell Wright scored 16, and Tyreek Duren crossed the 1,000th career-point mark, finishing with 14 points and six steals for La Salle (12-4, 2-1).
Following two Galloway free throws and a layup by Duren to put the Explorers up 69-65 with a minute left, the Flyers' Matt Derenbecker cut the lead to 69-67 on two free throws with 38 seconds remaining.
After two free throws from Galloway extended the lead to 71-67, Kevin Dillard answered for Dayton with a 3-pointer, making it a one-point game, 71-70, with 6 seconds remaining.
Tyrone Garland's free throw pushed La Salle's lead to 72-70 with 5 seconds left, and Dillard missed a late 3-pointer to seal the victory for La Salle.
Dyshawn Pierre and Vee Sanford scored 13 apiece for Dayton (10-7, 0-3).