The Falcons jumped to a 12-6 advantage and stayed ahead until 3:26, when Jaryd Eustace hit a 3-pointer and the RedHawks pulled ahead 25-23 and led 36-28 at halftime.
Bowling Green drew to within four at 12:36 in the second half after Jehvon Clarke drained two treys and Anthony Henderson hit another, but the RedHawks held off the charge and built a 10-point lead, then held on down the stretch.
The Falcons (9-10, 3-3) were hurt by 14 turnovers, which the RedHawks converted into 18 points.
Clarke led the Falcons with 22 and Henderson had 15. The loss snaps a three-game winning streak for the Falcons.