Knapke was 20 of 39 for 321 yards for the Falcons (4-2, 2-0 Mid-American Conference). Travis Greene and Fred Coppet each scored rushing touchdowns. Ronnie Moore had nine receptions for 178 yards and a touchdown and Tyler Tate kicked three field goals.
The back-and-forth contest had Buffalo (3-3, 1-1) leading 28-20 at the break.
A Knapke-to-Moore touchdown pass and a 31-yard field goal by Tate put the Falcons up 30-28 at the end of the third.
Buffalo's Anthone Taylor - who finished with 219 yards and three touchdowns - scored from 12 yards to give the Bulls a 35-30 fourth-quarter advantage.
But a 4-yard Coppet TD run with 2:19 left and a Falcons defense that held off a late Buffalo drive sealed the win for Bowling Green.