The Billikens (14-8, 5-3 Atlantic 10 Conference) trailed 65-63 with 2:12 left, and Reed's free throw pulled them within one with 1:50 to go.
Mitchell then put Saint Louis ahead 66-65 on two free throws with 55 seconds to play.
La Salle's Yves Mekongo missed a jumper and Steve Weingarten came down with the offensive rebound. Weingarten then had it stolen by Reed with 18 seconds left.
Mitchell hit two free throws with 17 seconds left for the final margin.
Saint Louis forced 14 Explorers' turnovers in the win.
Cody Ellis added 14 points as the Billikens snapped a two-game losing streak.