The Germans drew 0-0 in Wales but secured their passage thanks to Iceland’s 1-0 win over Denmark.
The Netherlands needed a 95th minute goal from Damaris Egurrola to see them to a 4-0 win over Belgium which meant they edged England on goal difference in Group A1 after the World Cup runners-up hammered the Scots in Glasgow.
The Dutch and the Germans will join France and world champions Spain in the semi-finals which take place in February.
The two finalists will qualify for the 2024 Olympics. If France, who are guaranteed a place as hosts of Paris 2024, reach the final, the team winning the third-place play-off will also progress.
England, who were hopeful of gaining an Olympic place for a Great Britain team which would have included their Scottish opponents, and the Netherlands went into the final round of games level on points but with the Dutch holding a two-goal advantage.
England quickly laid on the pressure, with Alex Greenwood and Beth Mead bracketing two goals in two minutes from Lauren James to give them a 4-0 half-time lead.
The Dutch, meanwhile, were only leading 1-0 through a 34th minute goal from Lineth Beerensteyn.
The English compounded their position when Fran Kirby added a fifth four minutes into the second half.
At that point, England had a two-goal advantage over the Dutch, but they then ran into some determined defence from the Scots who also went close, forcing a fine save from Mary Earps.
Beerensteyn added a second to give the Dutch hope, but the evening changed dramatically in stoppage time.
First, Egurrola found the back of the net to give the Dutch the advantage. That was followed by Lucy Bronze scoring a sixth for England but, even as they were celebrating, Egurrola snatched a second in Tilburg to send the Netherlands through.
“(Coach) Sarina (Wiegman) made sure we knew what was happening, and we knew what we were supposed to do,” said England forward Mead.
“We got the goals late on, we thought we had maybe gotten over the line but unfortunately it wasn’t meant to be.”
The Germans were level on 12 points with Denmark in Group A3 going into the game although their better head-to-head record meant they simply needed to match the Danes’ result against Iceland.
In the end, the Danish fell short with Karolina Vilhjalmsdottir, who plays in the Bundesliga for Bayer Leverkusen, where she is on loan from Bayern Munich, striking in the 77th minute in Viborg to give Iceland a 1-0 win.
The Germans struggled to get a grip on their game against Wales and in the end were fortunate to come away with a point.
“I thought we were the better team. We had the better chances and hit the post so, overall, I’m very pleased,” Wales manager Gemma Grainger told the BBC.
“It’s a real moment when you say we had the better chances against Germany.”
Wales, however, finished bottom of the group with this single point and are relegated to Group B.
World champions Spain had already qualified for the finals as Group A4 winners and were missing the injured Alexia Putellas, which may account for their sleepy start against Sweden.
Julia Zigiotti Olme put the visitors ahead in the first minute after non-existent defending at a corner.
Salma Paralluelo levelled 10 minutes later but more slack defence allowed Kosovare Asllani and Stina Blackstenius to put Sweden 3-1 up after half an hour.
Athenea del Castillo pulled one back for Spain who then turned the game around with three goals in the last 12 minutes, two from Mariona Caldentey, the other from Fiamma Benitez, to win 5-3.
Group A2 winners France, who had also booked their place in the finals, earned a hard-fought 1-0 win in Portugal, Grace Geyoro with the winner in the third minute of additional time at the end of the game.
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