The Bison, who were third in rushing defense last season in the Football Championship Subdivision, held the Red Foxes (1-1) to 57 rushing yards on 29 carries.
Wesley, who completed 20 of 27 passes for 190 yards with one interception, threw for scores of 6 yards to Travis Friend and 7 yards to Jake Flaherty, and Bucknell led 16-10 at halftime.
Sean Cobelli added a 36-yard fourth-quarter field goal for a 19-10 lead with 4:53 remaining after a 14-play Bucknell drive that took 7:29 off the clock.
Marist's Chuckie Looney, who completed 25 of 39 for 309 yards, threw an 11-yard touchdown pass to Matt Tralli with 31 seconds left, but Bucknell recovered the ensuing onside kick.
The game started 68 minutes late after a thunderstorm delayed pregame warm-ups.