Davis hit a 3-pointer for the first points in overtime and Smith added three free throws, the final two with four seconds left before the Billikens put in a layup at the buzzer.
Smith's layup with 36 seconds left in regulation tied the game at 46-all.
The victory came after Dayton (22-5, 12-3 Atlantic 10) had lost two games in a row for the first time this season and dropped just outside the Top 25. The win put them within a half-game of first-place VCU and a half-game in front of Saint Joseph's, both idle Tuesday.
The Billikens (10-17, 5-10) led 19-14 at halftime, the second-fewest points in a half in Flyers history. Milik Yarbrough led Saint Louis with 16 points.
Dayton finished shooting 32 percent. Saint Louis shot 28 percent and made just 1 of 8 shots in overtime.
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