The Golden Flashes (2-6, 1/3 Mid-American Conference) led 10-6 late in the third quarter when the Falcons' Eugene Cooper fumbled a punt return in Bowling Green territory.
Spencer then found Thompson two plays later for a 17-6 lead with 14:03 remaining.
The muffed punt was one of several miscues by the Falcons (4-5, 2-3) on offense and special teams. Bowling Green turned it over four times, had two kickoffs go out of bounds and missed a pair of field goals.
Kent State's Roosevelt Nix returned the Falcons' second fumble 9 yards for the final margin.
Keith finished with 179 yards on 16-of-23 passing as Kent State snapped a three-game losing streak and posted back-to-back wins over Bowling Green for the first time since 1972-73.