Dudzinski was 10-for-14 shooting and made all nine free-throw attempts while Burrell, a 5-foot-9 sophomore, converted 9 of 13 shots from the floor and 9 of 10 free throws.
Holy Cross (8-7) shot 51 percent (26 of 51) from the field, made 20 of 22 free throws and hit a season-high 55 percent (6 of 11) from 3-point range.
There were five ties and 10 lead changes before Cullen Hamilton broke open a 25-all game with a free throw. Burrell then scored Holy Cross' last eight points of the half to put the Crusaders up 34-25 lead at intermission.
Grant Mullins had 16 points for Columbia (8-6), which never got closer than eight after the break.
Mike Brey is looking at former Notre Dame players to fill two assistant jobs that have opened up with the departures of Martin Ingelsby and Anthony Solomon.
Purdue's 2016-17 prospects got a jolt Wednesday when forward Caleb Swanigan announced he will return for his sophomore season.
Clemson's Jaron Blossomgame says he is withdrawing from the NBA draft and returning to school for a final season.