A misplaced pass by France handed Argentina possession, which it swiftly turned into a try on the counter-attack.
Felipe Contempomi then converted to give Argentina the three-point victory.
France had started the brighter side, going ahead through a Morgan Parra penalty kick in the 3rd minute.
Argentina's Belisario Agulla scored a try one minute later after a poor pass from France number 10 Francois Trinh Duc gifted him an unchallenged passage to the try line.
Both sides traded penalty kicks, before France scored an imaginative try in the 26th with the score at 10-9 to Argentina.
A diagonal kick into the corner was knocked down by winger Yoann Huget, for number-8 Louis Picamoles to pick up and drive over the line.
Parra scored two penalties early in the second half to take France to a 20-13 lead, but Argentina responded through a penalty to trail by four. Contepomi then missed what appeared an easy penalty attempt in the 64th and France looked to be in control, before Montero stole victory in the final minutes.
France coach Philippe Saint-Andrea left the experienced trio of Thierry Dusautoir, Imanol Harinordoquy and Nicolas Mas at home with eye on the 2015 World Cup.
Argentina coach Santiago Phelan is preparing his team ahead of Los Pumas' debut in the Rugby Championship in August. Regular prop Rodrigo Roncero wasn't selected for Argentina.
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Scorers:
Argentina 23 (Belisario Agulla, Manuel Montero tries, Felipe Contepomi 2 conversions, 3 penalties) France 20 (Louis Picamoles try; Morgan Parra 5 penalties). HT: 13-14
-AP