Denmark will face Norway in the semi-finals on Thursday after edging France on penalties at the end of a tight contest in Linkoping.
Few were backing the Danes to prevail, but history repeated itself in agonising fashion for Les Bleues after Louisa Necib's penalty had cancelled out Johanna Rasmussen's opener in normal time. France lost a quarter-final on spot kicks to the Netherlands at Finland 2009, and after Necib had her attempt saved by the excellent Stina Petersen, they were heading for similar heartbreak. Sarah Bouhaddi kept out Theresa Nielsen, but Sabrina Delannoy hit the woodwork and Janni Arnth did the rest.
Élise Bussaglia swerved an effort wide to announce France's presence as the teams settled, but there was no lack of energy and commitment from their opponents, who were clearly going to have their say in the contest. Switching from 4-3-3 to 4-4-2 to tighten up their midfield, Kenneth Heiner-Møller's charges looked anything but overawed, and laid down a capable passing game to back up their ambitions.
It was Petersen who had to be the most alert of the goalkeepers early on, but Bruno Bini must surely have hoped for a more incisive start – and his side's frustration turned to angst when Rasmussen raced onto Katrine Søndergaard Pedersen's long ball over the top and blasted beyond Sarah Bouhaddi.
Trailing for the first time at Sweden 2013, France almost responded immediately when Gaëtane Thiney swivelled in the area and forced Petersen to save low to her right. Camille Abily then headed over and Denmark's impressive No1 batted away powerful attempts from Eugénie Le Sommer and Bussaglia, with her team-mates working similarly hard to keep France at bay.
Sensing the need to inject new ideas, Bini brought on Élodie Thomis for Sandrine Soubeyrand after the restart, yet there was no instant upgrade in the quality of France's openings. Instead, after Denmark's Pernille Harder had missed the target, both Necib and Abily took aim from distance and forced Petersen into stops.
It took Line Røddik bringing down Abily in the area for Les Bleues to really threaten the status quo, in fact, and for once Petersen had no answer. Her tournament had begun with a pair of penalty saves to thwart Sweden, but this time she could only help the ball into the net, Necib finally rescuing her side from the spot.
That remained France's high point after 90 minutes, and there was little for them to get excited about in the first period of extra time either, until Thiney drove narrowly off-target. Denmark, meanwhile, were struggling to get out of their half, and they were given another major scare when Abily curled a free-kick against the crossbar. France were desperate to make their domination count, but the winner never came – and their exit swiftly followed.
Player of the match: Christina Ørntoft (Denmark)