Roger Martinez and Duvan Zapata scored in the final 20 minutes for Colombia to give them their first victory over Argentina for 12 years.
Argentina, with both Lionel Messi and Sergio Aguero featuring, had dominated for much of the match as they aim to win the tournament for the first time since 1993.
Messi 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."
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