Achraf Yacoubou added 10 points for the Billikens (8-4) including two free throws with 11 seconds left to make it 58-50.
Vermont (5-6) closed on an 11-2 run but its last five points came after Yacoubou's free throws and were no threat to the outcome.
Yacoubou had the final six points of an 8-0 run that put Saint Louis up 56-44 with 49 seconds left.
Dre Wills's layup with 12:03 left gave Vermont its third and final lead of the second half at 36-35. Mike Crawford and Austin Gillmann followed with back-to-back 3s to give Saint Louis the lead for good.