Burdisso could be out for up to eight months
The centre-back damaged a posterior left knee ligament and tibia bone making a dangerous, late tackle on Colombia's James Rodriguez after half an hour of the match in the Caribbean coast city of Barranquilla.
"The defender has a lesion in the left knee and damage to the left tibia. Doctor Daniel Martinez remained in Colombia with the player and [on Wednesday] they will leave for Rome," the Argentina Football Association said on its website.
There was no mention of an estimated recovery period but the Argentine sports daily Ole said it could be from six and eight months.
Argentina, with Estudiantes defender Leandro Desabato replacing Burdisso, came from a goal down at half-time to beat the Colombians with second half strikes from Lionel Messi and Sergio Aguero in the final half hour.
After four matches, Argentina are equal with Uruguay, who have a game in hand, on seven points at the top of the South American group.