Sheehan went 28 of 41 for 376 yards for the Falcons (3-4, 2-1 Mid-American). Barnes' touchdown receptions were part of a 10-catch, 160-yard day. Chris Wright added six receptions for 104 yards for Bowling Green, which led 24-0 at the half.
Chris Bullock scored on a 26-yard run for the Falcons, and Jerry Phillips added a field goal.
Eric Williams rushed nine times for 88 yards and two touchdowns for the Cardinals (0-7, 0-3), and Steven Schott kicked a field goal. MiQuale Lewis carried 18 times for 115 yards.