Australia now go to Sydney on Wednesday to face the winners of the final last-eight tie between England and Colombia. Cortnee Vine scored the winning penalty to end a remarkable shoot-out that saw both teams take 10 spot-kicks, the quarter-final having ended 0-0 after 120 nerve-shredding minutes.
Vine held her nerve to send the crowd into raptures and keep the Matildas' dream of winning the World Cup on home soil alive.
Australia goalkeeper Mackenzie Arnold missed a chance to win the shoot-out when her kick hit the post, but then saved twice from Kenza Dali after the VAR spotted she had both feet off her line the first time.
Vicki Becho missed France's 10th penalty, and it was left to Vine to take Australia through by beating France's substitute goalkeeper Solene Durand, who had been sent on especially for the shoot-out.