South Africa, who qualified for the Games by winning the final repechage slot last month and lost 17-5 to New Zealand in Wednesday’s pool game, started strongly and defended brilliantly as the Tokyo silver medallists poured on the pressure late on.
South African captain Selwyn Davies picked a terrific line to blast through for the opening try then delivered a great kick and chase which he scooped up to gift Tristan Leyds a second try, both converted by Leyds.
New Zealand struggled for accuracy but put together a sharp move to send Moses Leo over to make it 14-7 at half-time.
New Zealand, who scored a last-gasp try to beat Ireland and top their pool earlier on Thursday, hammered at the door throughout the second half but were kept at bay by some incredible South African defence as they held on for a memorable victory.
The clash between Argentina, who won the regular season World Series, and France, champions of the new end-of-season finale, was always likely to be high-octane. But there was an extra edge with the Argentines being booed by French fans in the wake of a video showing some Argentine international footballers singing a racist song about the French team.
France got off to a flyer with two tries in a minute, as Andy Timo stepped inside his man for the opener before high-speed Aaron Grandidier Nkanang finished off another slick exchange.
Grandidier Nkanang then scored a spectacular diving one-handed try to make it 21-0 at halftime.
World player of the year Rodrigo Isgro, making his first appearance of the tournament after serving a suspension for foul play against France in the Madrid Sevens last month, got Argentina on the board.
A second Argentine try by Marcos Moneta set up a tense finale but the stadium exploded when replacement Antoine Dupont burst clear to settle it with France's fourth try in the final play.
Fiji extended their proud record of winning every match they have played at the Olympics since sevens was introduced in 2016 but they were pushed all the way by a remarkable Irish side who are relative newcomers to the format.
Iosefo Baleiwairiki claimed the first Fiji try after 30 seconds but Ireland led 10-7 at half-time after two well-finished tries by Chay Mullins.
Zac Ward then squeezed in for a third Irish try and a huge shock looked on the cards, but a typical length-of-the-field score finished by Joji Nasova and an opportunist try by Waisea Nacuqu after Ireland failed to gather the restart suddenly had Fiji four points ahead and they then defended superbly to hang on.
Australia, beaten at the quarter-final stage in the last two Olympics, were always on top against the United States and led 10-0 at the break after tries by James Turner and Corey Toole.
Maurice Longbottom then took centre stage, kicking a rare drop-goal penalty and then finishing the game with a try to complete a dominant 18-0 victory.
The medal matches will take place on Saturday.