Match ends, France 1, USA 2.
Women's World Cup: France 1-2 USA
Holders the United States overcame hosts France to book a semi-final date with England at the Women's World Cup.
Megan Rapinoe scored a goal in each half for the USA, the first a low free-kick that went in through a crowd of players, the second a side-foot finish.
France did not manage a single shot on target in the first half but Wendie Renard's late header gave them hope.
Les Bleues then had appeals for a penalty waved away after Amel Majri's cross struck Kelley O'Hara's arm.
The United States, who have never failed to reach the Women's World Cup semi-finals, will face Phil Neville's Lionesses in Lyon on Tuesday (20:00 BST).
The result also means there will be a Great Britain women's team at Tokyo 2020 because England are guaranteed to be one of the top three European teams at the World Cup.
Rapinoe brushes off Trump remarks to inspire USA
US president Donald Trump had criticised Rapinoe on Twitter before the quarter-final, telling the US women's team co-captain not to "disrespect our country" after she said she would not visit the White House if they won the World Cup.
Trump said Rapinoe, who scored two penalties as the US beat Spain in the last 16, should "win before she talks".
If the forward was affected by the episode, she showed no sign of it as she inspired her country to yet another World Cup semi-final.
Her opening goal came during a blistering start to the game, which saw three US attempts on target in the opening 15 minutes.
Rapinoe's angled free-kick after five minutes went through the legs of France captain Amandine Henry and into the net and the USA went on to control the half.
They doubled the lead when Alex Morgan's sublime pass allowed Tobin Heath to find an unmarked Rapinoe to score her fifth goal of the tournament.
The USA wobbled late on as Renard lifted the partisan crowd's spirits but the holders' vast experience saw them over the line - and to a meeting with the Lionesses.
A nation's dream ends where it began
This was billed as the biggest game in the history of French women's football.
In the end, a nation's dream ended in front of a 45,595 crowd at the Parc des Princes three weeks after it began at the same venue with a 4-0 drubbing of South Korea.
The hosts struggled before half-time in a game played in stifling heat in the French capital and did not produce their best football until they were 2-0 behind.
Kadidiatou Diani, one of their most dangerous players at this tournament, was kept quiet by Crystal Dunn, and it required a fine double save by Sarah Bouhaddi to frustrate Sam Mewis and Heath as the USA threatened to extend their lead at the start of the second half.
Renard's goal, a glancing header from a free-kick, gave France new belief and they thought they had a lifeline when Majri's cross struck the side of O'Hara's arm but referee Kateryna Monzul decided - correctly, replays suggested - that the defender's arm was not, in the language of the new handball law, making her body unnaturally larger.
Do England have anything to fear?
The United States and England met as recently as March, the Lionesses scoring two excellent goals through Steph Houghton and Nikita Parris in a creditable 2-2 draw at the SheBelieves Cup.
However, the USA will start as favourites at the 59,000-capacity Parc Olympique Lyonnais next week after demonstrating against France why they are the dominant force in women's football.
They have not been beaten at a World Cup since 2011 and rarely looked like losing in Paris, where they were on top despite having 10 fewer shots than France.
England will need to play their best football under Neville if they are to avoid another semi-final disappointment and reach a first ever Women's World Cup final.
Player of the match - Crystal Dunn (USA)
'Most intense match I've ever been part of' - what they said
United States boss Jill Ellis: "That was the most intense match I have ever been a part of. France are an incredible team. The surge from the fans was intense, like a tsunami."
France boss Corinne Diacre: "We're a long way off our target but I hope we have retained something elsewhere, and I hope we have won over the hearts and minds of the public and helped the women's game go further and move into that next level."
United States captain Alex Morgan: "We lived through a couple of storms during the match. I think we finished things off really well at the end. We knew there would be tough times and adversity, but we faced it straight on and we're moving on."
Heads you lose - the stats
- France have scored three headed goals at this World Cup, tied with the United States, Germany and the Netherlands for the most of any team.
- The USA have reached the semi-final in all eight of their Women's World Cups.
- Six of France's 10 goals at the 2019 Women's World Cup were scored via set-pieces, four of them scored by Wendie Renard.
- Since her debut at the Women's World Cup in 2011, Megan Rapinoe has been involved in more goals at the tournament than any other player (14 - eight goals, six assists).
- The host nation has been eliminated at the quarter-final stage in the past four Women's World Cups - China in 2007, Germany in 2011, Canada in 2015 and France in 2019 have all fallen at this stage.
- 19Mbock BathyBooked at 4mins
- 15BussagliaBooked at 90mins
- 9Le SommerSubstituted forAsseyiat 82'minutes
- 13GauvinSubstituted forCascarinoat 76'minutes
- 16LavelleSubstituted forHoranat 63'minutes
- 3MewisSubstituted forLloydat 82'minutes
- 15RapinoeSubstituted forPressat 87'minutes
- Kateryna Monzul
- Shots on Target
Second Half ends, France 1, USA 2.
Foul by Amandine Henry (France).
Tobin Heath (USA) wins a free kick on the right wing.
Élise Bussaglia (France) is shown the yellow card for a bad foul.
Foul by Élise Bussaglia (France).
Alex Morgan (USA) wins a free kick on the right wing.
Offside, France. Gaëtane Thiney tries a through ball, but Viviane Asseyi is caught offside.
Foul by Kadidiatou Diani (France).
Crystal Dunn (USA) wins a free kick on the left wing.
Substitution, USA. Christen Press replaces Megan Rapinoe.
Attempt missed. Kadidiatou Diani (France) right footed shot from the right side of the six yard box is close, but misses to the right. Assisted by Gaëtane Thiney following a set piece situation.
Amel Majri (France) wins a free kick on the left wing.
Foul by Tobin Heath (USA).
Attempt blocked. Élise Bussaglia (France) right footed shot from outside the box is blocked.
Hand ball by Megan Rapinoe (USA).
Attempt missed. Amandine Henry (France) header from the centre of the box misses to the left. Assisted by Amel Majri with a cross following a corner.
Corner, France. Conceded by Crystal Dunn.
Substitution, USA. Carli Lloyd replaces Samantha Mewis.
Substitution, France. Viviane Asseyi replaces Eugénie Le Sommer.
Goal! France 1, USA 2. Wendie Renard (France) header from the centre of the box to the high centre of the goal. Assisted by Gaëtane Thiney following a set piece situation.
Amandine Henry (France) wins a free kick on the left wing.
Foul by Kelley O'Hara (USA).
Corner, France. Conceded by Alyssa Naeher.
Attempt saved. Eugénie Le Sommer (France) right footed shot from the centre of the box is saved in the top centre of the goal. Assisted by Marion Torrent.
Attempt saved. Amandine Henry (France) right footed shot from outside the box is saved in the bottom left corner.
Delphine Cascarino (France) wins a free kick on the right wing.
Foul by Megan Rapinoe (USA).
Substitution, France. Delphine Cascarino replaces Valérie Gauvin.
Offside, USA. Alex Morgan tries a through ball, but Crystal Dunn is caught offside.
Attempt saved. Eugénie Le Sommer (France) header from the centre of the box is saved in the bottom right corner. Assisted by Amel Majri with a cross.
Corner, France. Conceded by Samantha Mewis.
Marion Torrent (France) wins a free kick in the defensive half.
Foul by Alex Morgan (USA).
Offside, France. Marion Torrent tries a through ball, but Valérie Gauvin is caught offside.
Goal! France 0, USA 2. Megan Rapinoe (USA) right footed shot from the centre of the box to the bottom left corner. Assisted by Tobin Heath.
Attempt saved. Valérie Gauvin (France) header from the centre of the box is saved in the top left corner. Assisted by Eugénie Le Sommer.
Delay over. They are ready to continue.
Substitution, USA. Lindsey Horan replaces Rose Lavelle.
Delay in match because of an injury Kelley O'Hara (USA).