The Bobcats (13-3, 1-1) never trailed, jumping out to a 12-0 lead and taking a 31-25 advantage into halftime.
The Bulls (8-5, 1-1) got to 33-30 with 15:07 remaining, then Ohio pulled away with an 11-0 spurt, capped by a Cooper three-point play.
Neither team shot well, with Buffalo (8-5, 1-1) outshooting the Bobcats from the field 38 percent (19 of 50) to 32.2 percent (19 of 59).
Ohio, which came in ranked sixth in the nation in turnover margin, scored 22 points off 23 Bulls turnovers while committing 15 miscues.
Mitchell Watt's 18 points led Buffalo, whose 52 points were a season low. Bulls leading scorer Javon McCrae (17.3 points per game), scored only eight points on 4-for-13 shooting.
The Bobcats lead the series 23-8.
Indiana guard James Blackmon Jr., who averaged 15.8 points in 13 games before being injured last season, has withdrawn from the NBA draft.
Isaiah Briscoe has decided to withdraw from the NBA draft and will return to Kentucky for his sophomore season. Marcus Lee intends to transfer.
The Big 12/SEC Challenge will again feature Kansas and Kentucky, this time at Rupp Arena. That's not the only exciting matchup in next season's midseason conference showdown.