The Purple Aces (8-2) scored 12 unanswered points in OT and made 23 of 28 free throws.
Ohio built a five-point second-half lead, 57-52, on a 3-pointer by Javarez Willis, but Evansville pulled ahead with a 13-2 run. Ohio rallied and Willis tied it at 69 with a 3-pointer to send the game into overtime.
The lead changed hands 11 times and there were six ties in the first half until Stevie Taylor hit a layup with 1 second to play for a 38-36 Ohio lead.
Maurice Daly Ndour led Ohio (3-4) with 14 points and 12 rebounds, Willis scored 12, and Antonio Campbell and Taylor had 10 apiece.