Chloe Kelly scored the winning penalty in Brisbane as European champions England beat Nigeria 4-2 in the shoot-out after a tense last-16 tie finished 0-0 at the end of extra time.
England survived a major scare before a crowd of 49,461, after coming through the whole of extra time a player short following the 87th-minute sending-off of James for a stamp on Nigeria's Michelle Alozie.
Georgia Stanway put the first penalty of the shoot-out wide for England, but Desire Oparanozie and Alozie followed suit, both missing the target with Nigeria's first two attempts.
Beth England, Rachel Daly and Alex Greenwood all netted for England before Kelly converted the decisive kick.
Sarina Wiegman's team, now the favourites to win a tournament which has already seen holders the United States, Germany and Olympic champions Canada go home, will play either Colombia or Jamaica in the quarter-finals in Sydney on Saturday.
"It was a very, very hard game, then we had a red card but we made it through extra time and then won on penalties. How we did that and how the team just kept going, I'm so proud of them," said Wiegman, who welcomed key midfielder Keira Walsh back into her side after injury.
"We keep pushing forward, there's more to come from this special team," added winning penalty-taker Kelly.
The defeat was cruel on Nigeria, the world's 40th-ranked team, who followed their impressive performances in the group stage by pushing England all the way.
Their England-born defender Ashleigh Plumptre crashed a shot off the woodwork in the first half, and Uchenna Kanu also hit the frame of the goal.
However, this was the third game out of four at this World Cup in which Nigeria failed to score.
"We had every opportunity to get the result and unfortunately we didn't get it done," said their American coach, Randy Waldrum.
"I think we could have (won it in 90 minutes). We had the best chances, we hit the crossbar twice in regulation play. We were a bit unlucky not to get something out of it.
"But credit to England, they are a great side."
England will discover the identity of their quarter-final opponents on Tuesday, when Colombia face surprise package Jamaica in Melbourne.
- Denmark take on co-hosts -
Australia faced Denmark in Monday's late game, which kicked off at 1030 GMT in front of a bumper crowd at Stadium Australia in Sydney.
A quarter-final against the winner of Tuesday's final last-16 tie between France and Morocco awaits.
Australia coach Tony Gustavsson had hinted that talismanic striker Sam Kerr would play a part against the Danes, although she began on the bench.
The Matildas captain has yet to see any minutes at the tournament after a calf injury she sustained before their opening game.
But she took a big step towards her recovery by resuming training with a ball on Saturday.
"We had a nice moment with the team yesterday to see her back with her boots on and touching the ball and being with the team in training," said Gustavsson on Sunday.
"It was a very good feeling for Sam, her teammates and me."
Australia beat Denmark 3-1 in a friendly last year, but Gustavsson was wary of the threat posed by a side that England found hard to break down in the group stage before winning 1-0.
"They have threats left and right and in very different ways," he said.
"It's going to be tough and we need to bring our A game."
Two-time defending champions the USA were dumped out in the last 16 on Sunday, going out of the Women's World Cup before the semi-finals for the first time as they lost 5-4 on penalties to Sweden after a 0-0 draw.
Sweden will face Japan in the quarter-finals on Friday, with the Netherlands taking on Spain the same day.
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