Roger Martinez and Duvan Zapata scored in the final 20 minutes after an Argentina side featuring Lionel Messi and Sergio Aguero had dominated.
Messi, 31, said the Group B defeat left his side “feeling bitter”.
“We didn’t want to start this way,” the five-time Ballon d’Or winner said.
“But now we have to lift our heads and keep going.”
Argentina have won the tournament 14 times – second only to Uruguay – but have not triumphed since 1993 and lost the last two finals on penalties against Chile.