After an early penalty from Morgan Parra, Belisario Agulla took advantage of Francois Trinh-Duc's wayward pass to cross and put the Pumas into a 7-3 lead.
But France stormed back and, thanks in part to a try from Louis Picamoles, Les Bleus led 14-13 at the break.
The sides traded penalties in a scrappy second half as France increased their lead to 20-16.
But Montero ran on to Joaquin Tuculet's kick in the closing minutes to give Argentina the win, helped by Felipe Contepomi's 13 points.
Argentina, who also beat Italy last weekend, will host France again in Tucuman next Saturday.
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The Barbarians are one of the top invitational rugby sides with a long history and classic rugby jersey.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Lions get ready for their matches with this green training jersey. It's what the players wear. Get in the Gear!
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The Lions are ready to get their Australia tour underway. They arrive in Perth on Monday.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.