Randolph threw for 422 yards and three touchdowns, including the game-winner to Koster with six seconds left. Matt Partain's extra point provided the winning margin.
Brett McDermott and Nick Cole caught touchdown passes from Randolph for Holy Cross (7-3, 5-0 Patriot League). McDermott had eight catches for 107 yards and Jon Brock added six catches for 122 yards for the Crusaders.
Lafayette (7-3, 3-2) held a 13-7 halftime lead and extended it to 20-7 early in the third on a 2-yard touchdown run by DeAndre' Morrow.
But the Crusaders responded with a 23-yard field goal by Partain later in the third and a 32-yarder early in the fourth. Cole's 22-yard touchdown catch tied the score at 20 with 5:36 left.
Morrow scored on a 22-yard run with 2:45 left, but a missed extra point kept the Leopard lead at 26-20 and gave Holy Cross the chance to win.
Morrow finished with 111 yards on 15 carries and Marc Quilling threw for 184 yards and a touchdown for Lafayette.
Holy Cross outgained the Leopards 490 total yards to 330.