The Bobcats (16-13, 8-6 Mid-American Conference) snapped a streak of 13 home victories for the Golden Flashes (19-10, 10-4).
Kent State scored the first five points in overtime, but Ohio took an 86-85 lead with 1:25 left on a three-point play by Ricardo Johnson, who was assisted by Cooper.
The Bobcats led 42-28 after a first half in which they outshot Kent State 56.7 percent (17 of 30) to 26.5 percent (9 of 34), but the Golden Flashes came back behind Rodriquez Sherman, who scored 14 of his 21 points in the second half.
Justin Greene had 19 points and 15 rebounds and Carlton Guyton had 18 points for Kent State.
Ivo Baltic had 16 points and Nick Kellogg 15 for Ohio.