The seventh-seeded Crusaders led 65-58 with 6 seconds to go, then Bryson Johnson and Darryl Shazier hit 3-pointers to pull the No. 2-seeded Bison within a point with 1 second left. Beinert made two free throws to seal the win.
Holy Cross (9-21), which entered the game having lost two straight and four of its last five, will play Lafayette on Sunday in the Patriot League semifinals.
Shazier scored 19 points and Johnson had 16 for the Bison, who had won eight of their previous 10 games.
The teams met for the eighth time in the last nine years in the tournament.
Holy Cross led 34-26 at halftime, but Bucknell opened the second half with a 7-0 run. The Crusaders also used a 7-0 run to break a tie at 51.