Les Bleus looked in total control at 20-3 after tries from Gael Fickou and Antoine Dupont, with 20-year-old fly-half Romain Ntamack converting both before kicking two penalties.
However, the Pumas hit back soon after the restart when Guido Petti Pagadizavalgh bundled over from a line-out, before Argentina replacement hooker Julian Montoya touched down.
Penalties from Benjamin Urdapilleta then put Argentina ahead, before Camille Lopez’s drop goal saw France edge through the closing stages.
Argentina had been aiming to repeat their shock victory over Les Bleus in the opening match of the 2007 World Cup in France, where they went on to reach the semi-finals.
After a cagey start, Nicolas Sanchez kicked a penalty wide in the 12th minute.
The Stade Francais fly-half, though, was soon off the mark when an offside in the French defensive line gave him another penalty opportunity in front of the posts.
France produced an instant response to score the opening try in the 16th minute.
Peato Vakatawa showed some great footwork before finding Fickou on a break down the left to go over. Ntamack, son of former France international Emile, made the conversion to put his side 7-3 in front.
Les Bleus scored another flowing try when scrum-half Dupont finished off an impressive move in the corner.
Ntamack added a penalty to extend the advantage to 17-3 as the half-hour mark approached.
After Argentina failed to make the most of some sustained pressure, fly-half Ntamack continued to show a cool head to convert another penalty and send France into the break with a seemingly healthy 20-3 lead.
Argentina, though, reduced the deficit just two minutes into the second half following a well-worked line-out.
A long, high throw saw second-row Pagadizaval bundled over. Sanchez converted the angled kick to bring Argentina back within 10 points.
Argentina replacement hooker Julian Montoya came on to win his 50th cap - and made an instant impact from a driving maul to score after Pagadizaval had taken a line-out catch in the 54th minute.
While Sanchez missed the conversion chance, the Pumas continued their momentum.
After a breakaway opportunity was eventually halted, replacement Urdapilleta kicked a penalty, for an earlier infringement by Damian Penaud, to close the gap down to just two points.
France were starting to lose composure and lock Sebastien Vahaamahina gave away 10 yards for a push, which put the penalty in kicking range.
After a TMO review for Emiliano Boffelli’s jumping challenge on Penaud, Castres Olympique fly-half Urdapilleta sent a left-foot effort over as Argentina edged in front 21-20.
France, though, produced an instant response when replacement Camille Lopez sent a drop-kick just over the bar, with Ntamack then missing a penalty chance.
There was still time for more drama as Urdapilleta saw his long-range kick drop just the wrong side of the posts as France held out - before tempers boiled over at full-time with some scuffles.
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