Cortnee Vine scored the decisive goal for the co-hosts after a record shootout in which the Matildas goalkeeper Mackenzie Arnold saved three French penalties. 

The 25-year-old from Victoria defied the pressure in front of a crowd of 49,461 at Lang Park - plus the millions watching on television - to calmly stroke home the penalty which sets up a semi-final next Wednesday (August 16) in Sydney against England, who beat Colombia 2-1 in today's other quarter-final. 

The match had finished 0-0 following normal and extra-time during which France thought they had clinched victory when Australia's Alana Kennedy put the ball through her own net following a corner only for the goal to be disallowed as Wendie Renard had committed a foul.

For the shootout, French coach Hervé Renard dramatically sent on substitute goalkeeper Solene Durand in place of Pauline Peyraud-Magnin in the belief she would save more of Australia's efforts.

Renard was proved right as Durand saved efforts from Steph Catley and Clare Hunt in the longest World Cup penalty shootout, men's or women's, as France and Australia duelled it out over 20 penalty kicks.

But when 19-year-old Vicki Bècho missed France's 10th penalty of the shootout and Vine scored, it meant France were knocked out at this stage of the World Cup for second successive tournament.

Afterwards, the Matildas did a lap of honour around the pitch as fans danced, waved flags and cheered a team which has captured the imagination in the sport-loving country.

Australia head coach Tony Gustavsson paid tribute to the supporters. 

"You are part of this win," he told them after Australia became the first host nation to reach the semi-finals of their home tournament since the United States 20 years ago. 

"You belong to this team tonight, every single person in this country."

France can consider themselves unfortunate to be going home but Les Bleues wasted the opportunity to make their dominance pay before the game went to penalties. 

"Tonight, we have to be proud of these girls who played an exceptional match," Renard said.

"It went from right to left, from left to right - to say who deserved it more is difficult.

"Congratulations to Australia, and congratulations to all the staff who have done a wonderful job."

England came from behind in the last quarter-finals of the tournament to beat Colombia in front of a crowd of 75,784 at Stadium Australia. 

The European champions are now the bookmakers' favourites to lift the World Cup trophy for the first time, despite having been far from their best so far.

England fell behind in the 44th minute when Leicy Santos put Colombia in front, but equalised in first-half injury time after Lauren Hemp tapped in from close-range following a bad error by goalkeeper Catalina Perez.

There was another slice of good fortune for England in the 63rd minute when a deflection gave Alessia Russo the chance to score only her second goal of the tournament.

This is the third consecutive World Cup that the Lionesses' have reached the semi-finals, having lost 2-1 to the US in France four years ago and 2-1 to Japan in Canada in 2015. 

England know they can expect to face a hostile atmosphere against the home team next week in the latest installment of a fierce sporting rivalry between the two countries that can be traced back to cricket in the mid-19th century. 

For now, England's Dutch coach Sarina Wiegman was just relieved to survive this latest test after England had scraped through the last round against Nigeria on penalties.

"These are big games and it has some physicality too - for them and for us," she said. 

"That's part of the game and we dealt with it really well.

"They got through it really well and got the win over the line. 

"That was what we were trying to do and fortunately we did that."

The first semi-final is due to take place at Eden Park in Auckland on Tuesday (August 15) between Spain and Sweden.