After a goalless 90 minutes, midfielder Groenen -- a former judo European Championship bronze medallist as a youth who recently agreed a move to Manchester United -- broke the deadlock in the 99th minute of a tense contest in Lyon, breaking Swedish hearts in the process.
When Danielle van de Donk's pass was touched into her path by Swedish defender Linda Sembrant, Groenen sent a low shot from 20 yards beyond the reach of goalkeeper Hedvig Lindahl and into the far corner.
Having knocked out former winners Japan and then Italy in the previous two rounds, the Dutch are now just one game away from adding a first World Cup to the title they won at Euro 2017, a tournament they hosted.
Getting this far is a remarkable achievement for the 'Oranje', who are appearing at just their second World Cup, but they will surely need to improve drastically if they are to stand any chance of defeating the holders.
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The USA are through to their third successive World Cup final after edging out England 2-1 on Tuesday.
That was a game which had it all, and it was always going to be difficult for this second semi-final to reach the same heights.
After the euphoria of ending a long hoodoo against old rivals Germany in the last round, 2016 Olympic finalists Sweden were the better team in the first half here.
Their coach Peter Gerhardsson was forced into a late change to his starting line-up, with midfielder Elin Rubensson initially dropped and then retained in the team after all as Nathalie Bjorn missed out with illness.
When Rubensson had a shot blocked in the area in the 37th minute, the ball fell to Lina Hurtig and her effort was well saved by Sari van Veenendaal.
That was about it for goalmouth action in the first half, and the Dutch goalkeeper and captain denied them again in the 56th minute as she got down low to tip Nilla Fischer's shot onto the post.
On this evidence, it is remarkable that Van Veenendaal was released by Arsenal prior to the tournament.
Then again, that was not the best save in the second half, with Sweden's Lindahl displaying stunning athleticism to tip a Vivianne Miedema header onto the bar soon after.
Extra time approached with increasing inevitability, although substitute Shanice van de Sanden -- a favourite with the locals as she plays for Lyon -- might have won it in stoppage time had her powerful strike not been tipped over by Lindahl.
The Dutch had seen Lieke Martens go off at half-time having been an injury doubt ahead of the game. They had lacked attacking inspiration for long spells but they finally got the breakthrough in the first period of extra time thanks to Groenen.
It was a fine goal by the 24-year-old, but it was agonising for Sweden, and the 2003 World Cup runners-up could not find an equaliser.
Instead they will face England in the third-place play-off in Nice on Saturday