The title-holders dominated the opening 90 minutes but a breakthrough proved elusive with Swedish goalkeeper Zecira Musovic making some startling saves to keep her side alive.
With the deadlock remaining after extra-time it went to penalties in Melbourne – and the tension went up another notch.
Sweden’s Nathalie Bjorn was the first to miss, skying high, only for Megan Rapinoe to do the same.
Rebecka Blomqvist’s effort was saved, again putting the US in the driving seat, only for Sophia Smith to blast wide.
When Kelley O’Hara hit the post, it was all down to Sweden’s Lina Hurtig, who made sure – but only after VAR ruled her kick had crossed the line.
The dramatic conclusion leaves the tournament without the two top-ranked sides after Germany’s group-stage departure, with the Swedes now considered among the chief frontrunners as the world number three.
Peter Gerhardsson’s side, who are striving to better their third-place finish in 2019, face the 2011 champions Japan next.
But it was a bitter blow for the US and an inglorious end for Rapinoe, the Golden Boot winner at the 2019 tournament. She will retire at the end of the season.
Expectations were high coming into the tournament for the four-time champions, who had made at least the semi-finals at every World Cup.
But they never looked up to the task in the group stage, with just one win, against Vietnam.
This devastating early exit will be seen as a major underachievement.
Musovic stands firm
Vlatko Andonovski made two changes from the team that drew 0-0 against Portugal, with Emily Sonnett in for suspended midfielder Rose Lavelle, and Trinity Rodman preferred up front to Lynn Williams.
Sweden rested nine players in their last group match, but reverted to their strongest 11 with strikers Fridolina Rolfo and Stina Blackstenius returning, and Musovic getting the nod as goalkeeper ahead of Jennifer Falk.
After both sides sized each other up, the US worked the first clear chance on six minutes, but Andi Sullivan’s drive from the edge of the box skewed wide.
A patient Sweden created space early on, particularly on the flanks, and sent in a series of teasing crosses to use their height advantage, but to no avail.
As the match wore on the US assumed midfield control and a livewire Rodman got close to getting the opener after 18 minutes, letting fly from distance with Musovic palming away.
She was forced into a similar stop minutes later, again from Rodman, but could only watch as a towering Lindsey Horan header rattled the crossbar in the best opportunity of a lively half the US dominated.
The title-holders carried the momentum into the second half and only a world-class reflex save from Musovic kept Sweden on level terms, turning a vicious strike from Horan around the post at full stretch.
It was all the US and a goal seemed inevitable, but Musovic again kept Sweden in the match with another cracking save in the 88th minute, this time from Alex Morgan.
She pulled off two more brilliant saves in extra-time to take the match to penalties.