Mackenzie Arnold made three saves before Cortnee Vine converted the winning spot-kick to seal the quarter-final victory in the longest shootout ever in the tournament's history in front of 49,461 screaming fans at Suncorp Stadium.
The Matildas will play either England or Colombia, who meet later on Saturday, in next Wednesday's semi-final at Stadium Australia in Sydney.
After France's Vicki Becho hit the post at 6-6, Vine coolly drilled her penalty into the bottom corner to send Suncorp Stadium into ecstasy.
Arnold had earlier slammed Australia's fifth penalty into the post but recovered to star in the 20-penalty shootout.
"I am still processing this. I think it might take me a couple of days but we have a quick turnaround so I have to figure it out quite soon. Unreal. I am speechless right now," Arnold said.
"I could have won the game for the girls but they rallied around me and kept me in it.
"At the end of the day it is my job to keep the ball out of the net and thankfully I could do that.
"Honestly we all do it for each other. They had my back 100 per cent."
That included denying Kenza Dali, being ruled to have come off her line, only to then save the midfielder's second attempt.
Caitlin Foord, Sam Kerr, Mary Fowler, Katrina Gorry, Tameka Yallop, Ellie Carpenter and Vine all converted their spot-kicks.
Steph Catley and Clare Hunt were both denied by French goalkeeper Solene Durand, who was substituted in for the shootout.
It followed a dramatic 120 minutes where both teams had chances to win it.
Hunt and Alanna Kennedy mostly kept Diani and Le Sommer quiet while Ellie Carpenter and Hayley Raso had an entertaining duel withwinger Selma Bacha.
France were clearly on top early as the Matildas lacked composure on the ball and consistently turned it over.
Kadidiatou Diani fired a warning shot in the eighth minute, Maelle Lakrar blasted a tap-in over the bar in the 12th minute while Arnold also denied Eugenie Le Sommer and Lakrar in the 28th and 32nd minutes.
But Fowler was denied by a heroic block from France right-back Elisa De Almedia in the 41st minute and goalkeeper Pauline Peyraud-Magnin four minutes later.
Kerr came on in the 55th minute and a minute later, burst forward and worked the ball to Raso, whose long-range effort was brilliantly denied by Peyraud-Magnin.
Neither team could find an opening from there and the game went into extra-time.
France skipper Wendie Renard headed home in the 100th minute but the goal was disallowed when it was ruled she had fouled Caitlin Foord in the build-up.
Vine toe-poked a difficult chance wide five minutes later while Arnold made a huge save to deny Vicki Becho in the 107th minute.
From there the rest was history.
