Fink's first scoring pass, a 30-yarder to Steve Tedesco, put the Pioneers (2-7, 2-5 Northeast Conference) ahead 10-7 early in the second quarter.
Marcel Archer gained 109 yards on 17 carries, including a 49-yarder and a 1-yard touchdown in the second quarter to boost Sacred Heart's lead to 17-7.
Michael Croce's 31-yard pass to Khalel Pritchard with 58 seconds left in the opening half made it 17-14.
Fink's 16-yard scoring toss to Rich Rossi, the only points in the second half, capped an eight-play, 73-yard drive and sealed the win.
Croce was 15-for-30 for 181 yards and a touchdown for Bryant (3-6, 2-4). Lindsey Gamble gained 73 yards on six carries for the Bulldogs.
Sacred Heart is 3-0 against Bryant, which it last faced in 2000.