The Bison (17-8, 8-1), who have won 15 of their last 17, led 42-31 with 18:21 to go.
After the Eagles (16-8, 6-3) pulled ahead on Vlad Moldoveanu's two free throws for 58-57 with 3:59 left, Bryson Johnson regained the lead for Bucknell with a 3-pointer with 3:30 left.
The Bison made 6 of 10 free throws down the stretch while American did not score again until Nick Hendra and Moldoveanu each made 1 of 2 foul shots in the final 8 seconds.
Troy Brewer led American with 13 points, while Moldoveanu, who was averaging 20 points per game, was held to nine.
The victory gave Bison coach Dave Paulsen his 300th career win. In his third year at Bucknell, Paulsen also coached at Williams, Le Moyne and St. Lawrence.