Brandon Wesley threw two touchdown passes as the Bison (2-0) built a 14-0 lead early in the second quarter. Wesley passed for 120 yards with touchdowns to Victor Walker and Frank DeNick.
The Bucknell defense kept Marist at bay, forcing three turnovers and holding the Red Foxes to 56 yards rushing on 25 carries.
It is the first two-game winning streak since 2009 for Bucknell, which leads the series 6-1.
Marist (1-1) cut the lead to 20-14 in the third quarter on a 3-yard touchdown pass from Chuckie Looney to Kevin Fitzpatrick. Bucknell put the game out of reach on an 8-yard run by Jeremiah Young with 3:42 remaining in the game.
Looney finished with 171 yards passing for the Red Foxes, but was intercepted twice and sacked three times.