The Bison (3-8) took a 17-14 third-quarter lead on Sean Cobelli's 32-yard field goal after Wesley was sacked on third-and-6 from the Bryant 7.
Joseph Francis' interception of a Mike Westerhaus pass and his 21-yard return early in the fourth quarter set up Bucknell at the Bryant 13, and Wesley found Tyler Smith for a 10-yard touchdown on third-and-7.
The Bulldogs (4-7), who had won four straight after an 0-6 start, cut their deficit to 24-21 on Westerhaus' 40-yard scoring pass to Jordan Harris with 7:41 remaining.
Harris was stopped short after a fourth-down catch on Bryant's next possession and Bucknell ran out the final 5:13.
Harris caught eight passes for 179 yards and two touchdowns, his fifth consecutive game over 100 yards.