Valerie Gauvin, who had earlier had a goal ruled out by VAR, put France ahead after a fine run by Kadidiatou Diani.

Brazil forced extra-time when Thaisa's drive, originally ruled out for offside, was given after another VAR check in Le Havre.

Amandine Henry got on the end of Amel Majri's long free-kick for the winner.

A tightly contested tie was minutes away from a penalty shootout when France captain Henry scored from close range to send Les Bleues into the last eight and a tie with either the United States - the 2015 winners - or Spain in Paris on Friday (20:00 BST).

• Relive the action from Le Havre as France reach quarter-finals

It was heartbreak for Brazil, who had come back strongly after falling behind to force one of the tournament's favourites to extra-time.

"I'm proud of our performance, the grit that we showed until the end," said Marta, Brazil's six-time world player of the year and scorer of a record 17 goals at World Cup finals.

"That's the feeling that I will keep with me. Those are the kind of matches that will help women's football to continue to grow."

Having finished behind Italy and Australia in Group C, the South Americans advanced to the knockout stage as one of the best third-placed teams.

Despite giving France a major scare, this was the end of the road for the 2007 runners-up even though the hosts again struggled to hit top form at Stade Oceane.

They did not register a shot on target in their previous match against Nigeria until the 65th minute.

Against Brazil, they failed to manage a shot on target in the first half after Gauvin's 23rd-minute header was ruled out after a lengthy VAR review.

Gauvin collided with keeper Barbara while challenging for Diani's cross, the ball ending up in the net.

Cheers turned to boos but home fans, who adopted Iceland's famous 'Thunderclap' in Le Havre, were celebrating when Diani set up Gauvin to slide France into the lead early in the second half.

Brazil's response was swift. Cristiane had a header tipped on to the bar before Thaisa equalised but only after another lengthy VAR check.

Eugenie Le Sommer wasted a fine chance to win it in normal time for France while Brazil lost Cristiane to injury early in extra-time.

In a thrilling conclusion, Griedge Mbock Bathy cleared Debinha's effort off the line before Henry pounced to ensure Les Bleues' World Cup journey rolls on to the Parc des Princes.

"We did what we needed to do tonight but it wasn't easy," said France coach Corinne Diacre. "Brazil were very tough and gave us a real run for our money, even in extra-time. Fortunately we were able to pull it out of the bag.

"We weren't at our best - I felt I didn't recognise my team out there. I told the players to start having fun out there, enjoy the game, and to be a bit more patient when they were on the ball. And that's what we saw in the second half.

"We're going to be watching the Spain-USA game very closely now and looking forward to the quarter-finals."

Brazil boss Vadao said: "The entire match was incredible. I think we've played four very good matches at this tournament and losing is just part of football.

"But we made it difficult for France. We equalised and had a chance to go ahead, but we were never going to boss the game for the entire match because these teams are the best in the world.

"Players like Marta and Cristiane should be playing for a few more years yet."

Another quarter-final for Les Bleues - the stats

• France have reached the quarter-final stages of the Women's World Cup in each of their past three appearances at the competition (2011, 2015 and 2019).
• Brazil have been eliminated at the last-16 stage in back-to-back Women's World Cups, and are winless in their past four knockout stage matches in the competition.
• Valerie Gauvin has scored seven goals in her past nine appearances across all competitions for France.
• Thaisa scored her first goal at the Women's World Cup, aged 30 years and 187 days - the only player to score her debut goal for Brazil in the competition at an older age was Sissi, who scored a hat-trick against Mexico in 1999, aged 32 years 17 days.
• Since making her debut in the Women's World Cup at the start of the 2015 tournament, no player has made more assists in the competition than France's Amel Majri.