Former champions Norway & Germany through after shoot-out, VAR drama
The 1995 champions, notably without Ballon d'Or winner Ada Hegerberg at this tournament, were clinical from the spot after Elise Kellond-Knight's late strike directly from a corner took the match to extra time and then penalties.
Norway had led after 31 minutes as Isabell Herlovsen kept her composure to finish, before a VAR review overturned what would have been a harsh penalty given against Maria Thorisdottir for handball.
The Matildas kept pushing for an equaliser and, after Sam Kerr saw an effort ruled out for offside, Kellond-Knight's unorthodox effort levelled the tie.
Norway peppered the Australia goal in the additional period, while Alanna Kennedy was sent off, but Vilde Risa's spectacular lob was as close as they came, bouncing behind off the crossbar.
Instead, the match went to penalties but Norway quickly took control as Kerr, Australia's star performer, blazed over the top.
Emily Gielnik then saw her tame effort parried away and Norway kept their cool, scoring through Caroline Graham Hansen, Guro Reiten, Maren Mjelde and Ingrid Engen to advance and play England or Cameroon next.
Earlier on Saturday, Germany defeated Nigeria 3-0 in Grenoble as the two-time champions continued their impressive form at the tournament.
Martina Voss-Tecklenburg's charges were the first team into the last eight, continuing their perfect record from the group stage and making it five clean sheets in a row.
Alexandra Popp's header and a Sara Daebritz penalty had them two goals ahead within 27 minutes, before Lea Schueller put the icing on the cake eight minutes from time.
VAR was again in the spotlight amid some long delays as Popp was ultimately allowed to stand despite Svenja Huth being stood in an offside position in front of goalkeeper Chiamaka Nnadozie, while it was also used in the penalty decision.
"There have been some strange situations of course," said Nigeria coach Thomas Dennerby
"There was one time in the first half when no one knows what they were watching for, players looking at each other, coaches looking at each other. I don't think anyone has the final solution for VAR yet."
Germany will face either Sweden or Canada, who play on Monday.