The Flyers (11-4, 1-0) rallied from an eight-point halftime deficit for a 64-61 lead with 28 seconds left in regulation after a pair of free throws by Dillard.
But Saint Louis wasn't done fighting, and Cody Ellis knocked down a 3-pointer with 2 seconds left to send the contest into overtime for the sixth time in the last 13 games in the all-time series.
In the extra frame, Williams stepped up from the foul line, going 4-for-4 in the first 1:40 for a lead Dayton would hold the rest of the way. Luke Fabrizius, in his first start, hit a game-high three 3-pointers for Dayton, including one in OT.
Brian Conklin led Saint Louis (12-3, 0-1) with 14 points.