The Falcons (3-0) downed Missouri, 51-28, last week and have won four in row dating to last season, their longest winning streak since 1994.
Bowling Green has posted consecutive 3-0 starts for the first time since 1963 and 1964.
Kansas (1-3) grabbed a 6-0 lead on Bill Whittemore's 16-yard pass to Marcellu Jones 1:34 into the second quarter. But the Falcons scored a pair of touchdowns in the final three minutes of the period to take a 22-9 halftime lead.
Linebacker Mitch Hewitt recovered a fumble in the end zone and Joe Alls tossed a 15-yard touchdown pass to Josh Harris with 1:51 left in the half.
Alls and Harris scored on runs of 11 and two yards in the first five minutes of the third period, extending Bowling Green's lead to 36-9.
Bowling Green nearly doubled Kansas in total offense, outgaining the Jayhawks, 503-277. The Falcons wore down Kansas with an effective running game that accounted for 350 yards.
Harris completed seven of 18 passes for 138 yards, but also rushed for 98 yards on 15 carries.
Whittemore completed 18 of 41 passes for 188 yards for the Jayhawks, who managed just 89 yards on the ground.