Reggie Keely added 10 points for the Bobcats (11-5, 2-0 Mid-American), who overcame 17 turnovers and held on after leading by 12 points midway through the second half.
Jared Klein led Western Michigan (8-7, 0-1) with a career-high 18 points. Shayne Whittington added 10 points and 13 rebounds for the Broncos, who shot just 28 percent from the field.
Ohio used a 12-2 run to build a 43-31 lead with 12:47 remaining, and still led by 10 with 8:35 to go.
A 19-6 Broncos run got them within 58-55 with 2:26 left, and a steal and layup by Klein with 7 seconds to go made it 60-59. Cooper hit 1 of 2 free throws for Ohio with 6 seconds left, and a 3-point attempt by Brown at the buzzer missed for Western Michigan.