Frohnapfel was 32 of 42 for 363 yards passing. He threw a 57-yard touchdown pass to Marken Michel early in the game, and connected with Michel again on an 5-yard scoring throw with 7:11 remaining to cap the scoring.
Tajae Sharpe had 15 catches for 159 yards with one touchdown for Massachusetts (1-3), which snapped a five-game losing streak dating back to last season.
Alex McGough threw a 1-yard touchdown pass to Thomas Owens for FIU (2-3) late in the second quarter. Massachusetts answered on the ensuing series when Logan Laurent kicked a 48-yard field goal to give the Minutemen a 17-7 halftime lead.
Alex Gardner bullied in from the 1-yard line with 20 seconds left in the third quarter to pull FIU to 17-14.
It was the first meeting between the schools.