In a madcap shootout on Sunday, Lina Hurtig struck the winning spot-kick, though US goalkeeper Alyssa Naeher parried the ball on the line and the result needed to be confirmed by VAR.
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Sophia Smith had had the chance to win the match for the four-time champions but blasted the ball over the bar.
Sweden goalkeeper Zecira Musovic emerged as the hero, having kept the Scandinavians in the match, making 11 saves before the shootout to deny the Americans.
Often starved of possession, the Swedes did not have a shot on goal until the 85th minute when substitute Sofia Jakobsson danced into the area, only to shoot straight at keeper Alyssa Naeher.
The talismanic Megan Rapinoe came off the bench to replace forward Alex Morgan in the ninth minute of extra time but was unable to create a winner for the Americans.
Renowned for having a dead-eye as a penalty shooter, Rapinoe struck the fourth kick for the US over the bar after Sweden’s Nathalie Bjorn had also blasted over.
Naeher saved Rebecka Blomqvist’s fourth shot for the Swedes but Smith’s failed effort kept the Scandinavians in it.
After Naeher and Magdalena Eriksson converted their spot-kicks, Kelley Ohara sent her shot pinging off the right post, allowing Hurtig to step up and send the champions out of the tournament.
Winners of the last two World Cups in France and Canada, the US slumped to their quickest exit, having made the semi-finals or better in all eight previous editions.
The Americans’ worst finish had been third place, three times.
Sweden will take on an impressive Japan in the quarter-finals on Friday.
The player of Sunday’s match was Musovic, who made 11 saves from the 22 US shots. Sweden had nine shots, with only one on target.
Musovic kept the match scoreless in normal time with two great second-half saves, including a diving stop to her right on an Alex Morgan header in the 89th minute.
In the 53rd minute, Musovic made a reactionary save with her left hand to deny Lindsey Horan’s shot that was targeted for the lower right corner.
Musovic made stops on Morgan and Lynn Williams in the first 10 minutes of extra time.
The lone save for Naeher was in the 85th minute from a 10-yard shot by Sofia Jakobsson.
Netherlands end South Africa’s run
In the last 16 game earlier on Sunday, the Netherlands secured a spot in the quarter-finals by ending South Africa’s historic run with a 2-0 win in Sydney.
The Dutch, runners-up in 2019, will now face Spain in the last eight but they suffered some anxious moments in front of 40,000 fans in Sydney and had goalkeeper Daphne van Domselaar to thank for making several important saves in the first half.
Jill Roord scored after nine minutes to put the Dutch on their way against a team that sits 45 places below them in the world rankings and was featuring in the knockout rounds for the first time.
South Africa goalkeeper Kaylin Swart then made a mess of an innocuous Lineth Beerensteyn shot midway through the second half to gift the Dutch a second.
The African champions will be kicking themselves after having numerous chances in the opening 45 minutes, almost all of them falling to their livewire captain Thembi Kgatlana.
“Daphne saved us a few times and I was happy to lead 1-0 at half-time,” said Dutch coach Andries Jonker.
“We did much better in the second half and had much more control of the game. We deserved to score again and, in the end, deserved to win.”
His team will now return to New Zealand for Friday’s last-eight game in Wellington against Spain, who outclassed Switzerland 5-1 on Saturday.
South Africa coach Desiree Ellis was proud but felt her side should have won.
“We had a lot of chances, especially in the first half, to kill off the game,” she said.
“The chances we created should have put us out of sight.”