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.
High school junior Zach Brown, a 7-footer from Miami, has rescinded his commitment to UConn, sources told ESPN.
Reserve center Akolda Manyang has been charged with felony aggravated assault after being accused of punching a man in the face without provocation.
ESPN college basketball reporter Andy Katz discusses the need for the Pac-12 putting in place rules on that would prohibit storming the court after a win.