Australia goalkeeper Mackenzie Arnold saved three French penalties and missed one herself in the shootout but it was teenager Vicki Becho hitting the post with the 12th spot-kick that opened the door for the hosts.
Cortnee Vine showed nerves of steel when she stepped up to fire the ball past Solene Durand into the bottom right corner of the net and send the Matildas into a semi-final against England or Colombia on Wednesday.
The substitute winger raced back towards the centre circle to celebrate with her team mates as the stadium and fan zones around the country exploded in delight.
Spain will play Sweden on Tuesday in Auckland for the other spot in the final.
France, who were looking to reach the semi-finals for the second time after 2011, had the better of the first half and extra time and will rue a few chances left out on the pitch as they head home.
Australia forward Mary Fowler had earlier spurned several chances -- one stopped by a sensational block from Elisa de Almeida -- and captain Sam Kerr was unable to get a clear shot on goal after being introduced as a 55th-minute substitute.
France dominated possession for much of the match and had the best of the early chances with Maelle Lakrar fluffing a golden opportunity to give Les Bleues the lead from a corner in the 12th minute.
Australia's only early chances came against the run of play from set pieces but Fowler had as many as six as the game opened up around halftime.
Kerr entered the fray to a huge ovation in the 55th minute, her first charge forward giving Hayley Raso space for a rasping long-range shot that France's starting goalkeeper Pauline Peyraud-Magnin did well to push over.
France weathered the storm triggered by the introduction of the Australia captain, however, and their 19-year-old substitute Becho was soon threatening down the other end.
The last quarter of an hour of normal time was a tense but ragged affair with half chances at both ends, a theme reprised in the first half of extra time.
Australia's Alanna Kennedy headed the ball into her own net in the 100th minute under the attentions of Wendy Renard only for the France captain to be penalised for a foul on Caitlin Foord.
France had the better of the second period with Becho lashing a shot from the edge of the box in the 107th minute that goalkeeper Arnold saved well.
That was just a precursor to her heroics in the shootout, in which she twice saved penalties from Kenza Dali with her first stop scratched off because she moved off the line before it was taken.
Alessia Russo of England celebrates her match-winning goal against Colombia in Sydney. Photo: Getty Images
England come back to beat Colombia
Alessia Russo's second-half goal lifted England to a thrilling 2-1 quarter-final victory over Colombia and into the Women's World Cup semi-finals for the third consecutive time.
Lauren Hemp also scored for the European champions, who now face hosts Australia on Wednesday in Sydney for a place in their first World Cup final.
"It's exciting, what more do you want in front of a home crowd?" Russo, who was named player of the match, told ITV. "We got a bit of that tonight against Colombia. You want to play against the best teams.
"England fans can always dare to dream but we'll carry on working hard," Russo said.
Colombia, who were the lowest-ranked team remaining at 25 and had never made it past the round of 16, opened the scoring with a goal from Leicy Santos to the delight of the large swathes of yellow-and-blue clad Colombian fans among the crowd of 75,784 at Stadium Australia.
The fourth-ranked Lionesses, missing top scorer Lauren James to a two-game suspension, were one of just four top-10 ranked teams left standing -- along with Sweden (3), Spain (6) and Australia (10) -- in a tournament that has been full of delightful twists and turns.
"I'm very, very happy to stay here for another week," coach Sarina Wiegman said. "I think the team showed again some resilience and a lot of togetherness."
Asked what Wiegman -- the lone female head coach remaining from the original dozen -- said in the post-game huddle, Russo answered: "Sarina said that she was proud. Lots to work on and we get rested and go again."
Russo netted the winner in the 63rd minute, pouncing on a through ball from Georgia Stanway that bounced off a Colombia defender and into the path of the Arsenal forward, who fired low and hard into the bottom corner.
Santos scored against the run of play in the 44th minute, launching a curling ball that deflected off the leg of Rachel Daly and into the top corner of the 18-yard box, grazing Mary Earps's fingertips and going into the far corner of the net.
Trailing for the first time in the tournament, the Lionesses, who are undefeated in 37 of their last 38 games, kept their composure and when Colombian keeper Catalina Perez fumbled and lost the ball on Russo's shot, Hemp was there to poke it in from six yards out just before the halftime whistle.
"This team is really special. Long may that continue," Hemp said. "The atmosphere was incredible. The connection I have with Alessia is really strong. Bring on Australia. I am absolutely buzzing. I am over the moon and can't wait."
Lorena Bedoya Durango had an excellent chance at an equaliser with a long-range effort late in regulation time that Earps deflected over the crossbar.
Eighteen-year-old Linda Caicedo, who was one of the most exciting players at the World Cup, had a last chance deep in extra time but shook her head after firing the ball over the bar.
