Off an inbounds pass, Ashman drove to within six feet of the basket, then drew two Bonnie defenders before calmly sinking the shot to send Dayton (11-16) to a Thursday matchup against George Washington.
St. Bonaventure (14-15) lost a chance to send the game into another overtime after David Messiah Capers missed a short jumper and Isaac King failed to convert an offensive rebound as time expired.
Tony Stanley led the Flyers with 16 points, while Capers had a game-high 18 for the Bonnies.
St. Bonaventure sent the game into overtime after Tim Winn grabbed an offensive rebound and hit a 12-foot baseline jumper with 27 seconds left in regulation to tie the game at 64.
Winn, the Bonnnies' leading scorer, was held to four points on 2-of-9 shooting.
Dayton had a chance to hold for the last shot in regulation, but Edwin Young threw an errant cross-court pass to Cain Doliboa that caused Doliboa to step out of bounds.
Behind 62-60 with 1:34 left, the Flyers came back to take a two-point lead behind a turnaround baseline jumper by Ashman and two Young free throws.
Dayton won despite getting outrebounded by St. Bonaventure 49-32. Caswell Cyrus grabbed 10 for the Bonnies, while scoring four of his 12 points in overtime.
Trailing 40-34 at halftime, St. Bonaventure came out on a 13-4 run to start the second half, capped by a King putback of a J.R. Bremer missed free throw that gave the Bonnies a 47-44 lead.
Dayton hit 2-of-8 from the field during that span, while Bremer had six points.
Dayton took their six-point halftime lead behind seven points each from Stanley, Doliboa, and Coby Turner.
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