Rapinoe, who was involved in a spat with U.S. President Donald Trump during the build up to the game , opened the scoring in the fifth minute with a low-struck free-kick from the left that went in untouched through a crowd of players.
The 33-year-old Rapinoe added their second goal in the 65th minute and although France got back in the game with a Wendie Renard header in the 81st, the U.S. held firm to maintain their record of never missing out on the World Cup semi-finals.
A night which showed just how far the women's tournament has grown, with thousands of travelling American fans in a sell-out 45,000 crowd, ended in disappointment for the hosts.
Corinne Diacre's French team played attractive, passing football but, while their style was more contemporary and aesthetically pleasing, it was the killer edge of the experienced Americans that proved decisive.
"You have to give it up to the French team. They outplayed us for sure with the ball today," said Rapinoe.
"But we were so good defensively, so strong. We hit them where it hurt, we took our chances. This team has an unreal amount of heart," she added.
Heart and brains too.
Seeing the French defence had put just two players in the defensive wall for the early free kick, Rapinoe accepted the invitation and her low drive flew through a crowded area and past unsighted goalkeeper Sarah Bouhaddi.
FEW OPENINGS
The U.S. have scored in the opening 12 minutes of every game in this tournament and they also could draw on the fact that they have never lost after taking the lead in a World Cup game.
France's hopes of turning the match around depended on them opening up the American defence with their clever passing but while they impressed in build-up play they created few openings.
France right winger Kadidiatou Diani put left back Crystal Dunn under intense pressure with her frequent dribbles down the flank, while on the left Amel Majri pushed progressively into an advanced role to the consternation of right back Kelley O'Hara.
Yet despite all that promising play in the wide areas, the French struggled to provide striker Eugenie Le Sommer with real quality service and, having survived a wave of pressure after the interval, Rapinoe struck again.
Alex Morgan's clever through ball put Tobin Heath into space on the right and her low cross, pulled back behind the French line, found the hovering Rapinoe who slotted home her fifth goal of the tournament with confidence.
Heath's effort that would have wrapped up the game was ruled out for offside, and France responded with one last push.
With nine minutes of normal time remaining, the home side finally got some reward when Renard rose above the American defence, whose marking was unusually sloppy, to glance in a header from a Gaetane Thiney free kick.
There was a scare for the Americans when a ball from Majri struck O'Hara's arm inside the area, but the Ukrainian referee waved away the appeal, despite it being the kind of infringement that has been penalised in other games at the tournament.
The Americans survived and while Ellis's side may not have the quality of passing and movement shown by the French, they have a resilience ingrained in their approach.
"That was the most intense match I have ever been a part of. A win’s a win in a World Cup. I haven’t seen many pretty games in a World Cup," said Ellis.
"France are an incredible team. The surge from the fans was intense, like a tsunami. I credit the players. They’ve taken everything on."
Next up for the holders are England on Tuesday in Lyon and that should prove an equally tough test.
For France, the inquest will begin in to how this talented side, once again fell short.
"I still have work to do," said Diacre. "There are still certain things we need to fine-tune and work on together. Yes, it is a failure on a footballing level. We’re a long way off our target but I hope we have gained something elsewhere, and I hope we have won over the hearts and minds of the public."
(Reporting by Simon Evans; Editing by Ken Ferris and Toby Davis)
