After a goalless 90 minutes, Dutch midfielder Jackie Groenen — a former judo European Championship bronze medallist as a youth — broke the deadlock in the 99th minute of a tense contest, breaking Swedish hearts in the process.
The shot went into the far corner, meaning there was nothing Swedish goalkeeper Hedvig Lindahl could do, despite an impressive performance throughout the match.
“We never knew this would be possible. It's one more match and we might possibly be world champions,” said a beaming Groenen.
The Netherlands have long been a force in the men's game, but they had never even been to a major women's tournament until a decade ago. They will now face the top ranked team worldwide in the final when they go up against the USA.
After ending a long hoodoo against old rivals Germany in the last round, 2016 Olympic finalists Sweden were the better team in the first half but lost momentum after the 90 minutes were over.
A Sari van Veenendaal save from Lina Hurtig in the 37th minute was the only chance of note in the first half, but the Dutch goalkeeper and captain denied Sweden again in the 56th minute as she tipped Nilla Fischer's low shot onto the post.
That was not the best save in the second half, though, with Sweden's Lindahl displaying stunning athleticism to tip a Vivianne Miedema header onto the bar.
Extra time approached, although substitute Shanice van de Sanden might have won it in stoppage time had her powerful strike not been tipped over by Lindahl.
The 2003 World Cup runners-up Sweden also saw midfielder Kosovare Asllani go off on a stretcher at the end after being hit in the face by the ball. She underwent CT and MRI scans later, but the team doctor said these showed “no signs of more serious damage”:
Sweden will face England in the third-place play-off in Nice on Saturday.
“Now it's about the medal,” said coach Peter Gerhardsson. “Ending up third of course feels a lot better than ending up fourth. We'll do everything in our power to try to do that.”