The two teams were contesting the third place play off at the Copa America following Messi's side elimination in the semi-finals against Brazil.
Chile were the losing side in their semi-final match against Peru.
Messi was controversially sent off in the first half after butting heads with Chilean midfileder Gary Medel, with some observers believing Messi to be the victim.
After the match, Messi appeared to claim that there was corruption in the Copa America favouring host side Brazil, and had already criticised both CONMEBOL and refereeing standards in the competition.
Fellow Argentine Nicolas Otamendi also failed to collect his medal.