Yann Sommer saved the final spot kick from Kylian Mbappé as Les Bleus were eliminated on penalties.
Switzerland beat France on penalties to progress to a quarter-final against Spain on an evening packed with drama in Bucharest.
Didier Deschamps set up with a three-man defence but his side struggled to adapt to the unfamiliar formation and found themselves behind when Haris Seferović's towering header beat Hugo Lloris. The lively Kylian Mbappé and Adrien Rabiot fizzed efforts wide but the Nati were good value for their half-time lead.
Kingsley Coman's introduction at the break instigated a change of shape for Les Bleus but it was the shift in their fortunes that proved more telling. Lloris altered the momentum, brilliantly saving Ricardo Rodríguez's penalty after Benjamin Pavard's foul on Steven Zuber, and France took full advantage.
Mbappé slipped in Karim Benzema to clip in the equaliser and the Real Madrid forward nodded in two minutes later after Yann Sommer had parried Antoine Griezmann's shot. Two became three courtesy of a stunning curling strike by Paul Pogba.
Switzerland pulled one back late through another Seferović header and, just when it seemed France had done enough to edge this last-16 thriller, Mario Gavranović rifled a dramatic equaliser from outside the box to force extra time.
Mbappé spurned the best chance of the additional 30 minutes and then, after nine successful spot kicks in the shoot-out, it was the Paris striker who was denied by Sommer to seal an unforgettable night for Vladimir Petković's Nati.
"Very impressive in the midfield, offensively and defensively. The leader of his team with great communication and organisation."
Corinne Diacre, UEFA Technical Observer
Check out every Heineken Star of the Match at UEFA EURO 2020.
David Crossan, France reporter
The pressure on Mbappé proved too much at the end but the shoot-out wasn't the real issue for France tonight. Lloris's penalty save looked like being the turning point and, after scoring three quick goals, there's no way the French should have allowed the Swiss back into the match. Switzerland played superbly and taking off Lenglet at half-time was an admission from Deschamps that he got his initial set-up wrong.
Vieri Capretta, Switzerland reporter
History is written! Switzerland win their first ever EURO knockout tie and they do it in incredible fashion, beating the world champions. It was the perfect night. The Swiss were solid at the back, high tempo in pressing and going forward, frustrating the France front three and able to break the deadlock. That penalty miss shifted the momentum but the reaction was unbelievable. Switzerland were always going to have to be perfect against France and, in the end, they were.
Reaction
Granit Xhaka, Switzerland captain: "It's amazing. We made history tonight and we are all very proud. We wrote the history of this footballing nation. Now we have Spain in the quarters. It will be difficult but we are dreaming now."
Yann Sommer, Switzerland goalkeeper: "What a match! What an evening of football! It was our chance to finally go through the round of 16 because we never made it before. It's incredible – we played with heart and with character. It was a really difficult situation for us after the penalty miss. I'm really proud of the team, how they came back. We always believed."
Hugo Lloris, France goalkeeper: "We're not looking for excuses. The regret, if we can have one, is that we led 3-1. In recent years we have known how to shut up shop. We went through every emotion possible and honestly that was football the way we like it. The two goals we conceded in the last quarter of an hour really hurt us."
Raphaël Varane, France defender: "It's hugely disappointing. We completely messed up our first half. We reacted in the second half but then left them space and they came back. Penalties are a lottery. We could have scored in extra time as we had the chances to do that."
Key stats
Pogba has now scored a goal in each of the last four major tournaments – 2014 and 2018 World Cups, EURO 2016 and 2020.
This was the seventh time Benzema has scored two goals in a game for France. He has yet to get a hat-trick.
This was the first time France had conceded three goals in a EURO knockout match since 1960 (4-5 versus Yugoslavia).
France have only kept one clean sheet in their last ten EURO final tournament knockout phase matches.
Switzerland had lost all three of their competitive penalty shoot-outs before tonight.
The Nati had scored eight goals across four EURO final tournaments prior to this year – they have netted seven so far at EURO 2020.
Seferović became the second Swiss player to score in consecutive EURO finals games after Hakan Yakin in 2008.
Switzerland have won all bar two of their matches when Seferović has scored for them (W16 D1 L1).
The last time Switzerland were successful in a final tournament knockout phase match before this game was back in the 1938 World Cup.
Line-ups
France: Lloris; Varane, Lenglet (Coman 46, Thuram 111), Kimpembe; Pavard, Pogba, Kanté, Rabiot; Griezmann (Sissoko 88); Mbappé, Benzema (Giroud 94)
Switzerland: Sommer; Elvedi, Akanji, Rodríguez (Mehmedi 87); Widmer (Mbabu 73), Freuler, Xhaka, Zuber (Fassnacht 79); Shaqiri (Gavranović 73); Embolo (Vargas 79), Seferović (Schär 97)
