A late try for Manuel Montero maintained Argentina's proud home record over the French and helped them claw back the advantage in the IRB World Rankings.
The French seemed in control with full-time looming but Montero's score, converted by Felipe Contepomi, snatched a 23-20 victory.
The result leaves France without a win in Argentina since 1998 and lifts the Pumas to sixth in the standings; France drop to seventh.
Argentina's first home game of the forthcoming Rugby Championship will be against South Africa in Mendoza, in the heart of the country's wine-making industry. France, who play the Pumas again in Tucuman on Saturday, can vouch for the quality of this Argentine vintage.
The thrilling conclusion in Cordoba was mirrored in the Antipodes, with the All Blacks winning 22-19 against Ireland in Christchurch and Australia beating Wales in Melbourne with the final kick of the match.
The results, plus England's 36-27 defeat by the Springboks in Johannesburg, mean the top three - New Zealand, Australia, South Africa - stay as they are.
However, there was some consolation for the Welsh as their two-point defeat let them seize fourth spot from England by virtue of losing to a higher-ranked team.
Wales thus become the highest-placed European side, while Argentina's ascent leaves the southern hemisphere with four of the top six in the world.
The weekend pitted eight of the top 10 sides against one another, which is particularly relevant with Rugby World Cup 2015 seeding due at the end of the year. Of the automatic qualifiers, the top four sides are guaranteed separate pools.
Samoa squeaked past Japan 27-26 in Tokyo on Sunday to seal the Pacific Nations Cup and consolidate ninth in the world.
Samoa's opponents in Apia this Saturday, Scotland, are rewarded for their 37-25 defeat of Fiji with 10th place.
Georgia drop one place to 15 after their 36-20 reverse against USA, who stay 17th in the IRB World Rankings.
Uruguay started the week with defeat by Russia - which lifts the Russians from 21st to 19th - but finished on a high note with a convincing 35-7 victory over Portugal. While that result does not alter their 22nd position, the Uruguayans are within half a ranking point of a top-20 spot.
After defeat the previous week by Argentina, Italy got its tour of the Americas back on track with a win over Canada to strengthen its 12th place by more than one ranking point.
Top 20 on the IRB rankings (last week's position in brackets):
1(1) New Zealand 91.43
2(2) Australia 86.77
3(3) South Africa 85.47
4(5) Wales 82.54
5(4) England 82.49
6(7) Argentina 81.22
7(6) France 81.06
8(8) Ireland 79.85
9(9) Samoa 77.64
10(11) Scotland 76.56
11(10) Tonga 76.19
12(12) Italy 75.26
13(13) Canada 71.74
14(15) Fiji 69.21
15(14) Georgia 68.52
16(16) Japan 67.93
17(17) United States 67.38
18(18) Romania 64.54
19(21) Russia 62.05
20(19) Spain 61.63