Players and coaches from both sides clashed on the pitch at the end as the hosts booked their last-four spot.
Crystal Palace striker Jean-Philippe Mateta got France’s winner when he headed home Michael Olise’s fifth-minute corner at the near post.
Argentina’s best chance came just before half-time with Giuliano Simeone heading over from close range.
France had a late goal ruled out as the video assistant referee (VAR) ruled Maghnes Akliouche had fouled an opponent before Olise found the net with a deflected effort.
There has been recent animosity between the teams, with Argentina players filmed singing a derogatory song about France’s black players in the aftermath of winning the Copa America on 14 July.
French fans booed the Argentinian national anthem before the match at Nouveau Stade de Bordeaux, and the fracas at the final whistle highlighted the rivalry and tension.
France, who last won Olympic gold in men’s football in 1984, will face Egypt in Monday’s semi-final (20:00 BST) after the Pharaohs beat Paraguay 5-4 on penalties following a 1-1 draw.
Argentina, who beat France on penalties at the 2022 World Cup, were hoping to become the first nation to be both the reigning world and Olympic champions since Italy in 1936.
The result rounded off a brilliant day for French sport with the 2024 hosts winning gold medals in judo, swimming and cycling BMX earlier on Friday.