Schilz finished 16 for 23 passing with 258 yards and an interception, leading the Falcons (2-0) to a 35-10 halftime lead with his prolific first-half passing.
Bowling Green outgained the Bears 572-193 in offensive yards, including a 304-37 margin through the air.
Schilz connected with Eugene Cooper on touchdown passes of 37, 6, 23 and 18 yards in the first half. Four of Cooper’s six receptions were for scores, as he finished with 134 yards.
Jordan Hopgood rushed 12 times for 90 yards and also had 37 yards and a touchdown on three receptions for the Falcons, who have won consecutive games to open a season for the first time since 2003.
Travis Davidson led the Bears (0-2) with 89 rushing yards and a touchdown.