Cooper, who missed a shot with 1 second left in regulation for the Bobcats (2-0), got them off to a quick start in the extra period by hitting a 3-pointer after 22 seconds had elapsed, and they led the rest of the way.
Ohio led 40-31 just 1:14 into the second half, but the Cardinals (1-2) tied it with a 13-4 run, the fifth of 10 ties in the game.
A three-point play by Cooper put Ohio ahead 71-67 with 1:30 left in regulation, but two free throws for Lamar by Osas Ebomwonyi and a layup with 24 seconds left by Charlie Harper tied it for the final time.
Stevie Taylor and Ivo Baltic scored 14 points each for Ohio.
Mike James led Lamar with 20 points and Harper had 18.