The Falcons (13-18, 8-8 Mid-American Conference) led 37-24 at halftime after holding the Bulls to just one field goal in the final 4:19 of the opening half and 8-of-26 shooting (30.8 percent) overall.
A layup from Scott Thomas with less than 6 minutes remaining gave Bowling Green its largest lead of the game, 62-45, but a 12-2 run cut the Buffalo deficit to 64-57 with 2:49 left to play.
Brown went 6 of 6 from the free-throw line in the final 1:24 to seal the win for the Falcons, who snapped a six-game losing streak and locked up the No. 7 seed in the conference tournament. The Falcons will host Northern Illinois on Tuesday.
Javon McCrea led the Bulls (17-12, 8-8) with 19 points.