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.
The Barcelona talisman missed their best chance to score, heading wide after David Ospina had saved from Manchester City defender Nicolas Otamendi.
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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Copa America 2019 – Neymar, Qatar and the Maracana
Aguero determined to help Argentina win Copa America for Messi’s sake
