The Falcons (3-2, 1-0 Mid-American) took the lead with 2:57 left in the first after Knapke threw a 27-yard TD pass to Ronnie Moore.
The lead changed hands six times and Bowling Green never held more than a two-score lead.
Knapke passed for 443 yards and finished with three interceptions. Greene ran for 110 yards on 20 carries.
Massachusetts climbed back from a 12-point deficit and took its last lead in the third quarter after 80- and 18-yard touchdown passes from Blake Frohnapfel.
Frohnapfel threw for a career-high 589 yards - smashing his previous high of 267 yards - and five touchdowns for the Minutemen (0-5, 0-1).
Massachusetts fumbled on its last possession and the Falcons recovered, running out the clock for the win.