The Tar Heels played without standouts P.J. Hairston and Leslie McDonald, who are sitting out while eligibility issues are being resolved by the school.
The game was tied at halftime and Holy Cross trailed by only six heading into the final minute. Paige hit four free throws down the stretch to help secure the victory, North Carolina's second in as many games this season.
Brice Johnson added 12 points for the Tar Heels and James Michael McAdoo had 11. North Carolina shot only 37 percent from the field.
The Crusaders (1-2) were led by 13 points by Malachi Alexander.