The stage, which began in Formigal and ended at the iconic Col du Tourmalet, featured three other climbs before the final ascent to the finish.
The relentless climbing and downhill racing took its toll on many, including last year's winner Evenepoel, who now finds himself out of contention.
It was the Jumbo-Visma team which controlled the stage, with Sepp Kuss (28) in the red jersey, Primoz Roglic (34) and Vingegaard all in the leading group on the final climb, before Vingegaard made his telling break with eight kilometres to the finish.
I'm just so happy and I couldn't choose a better day; today is my daughter's birthday and I wanted to win for her so badly today," Vingegaard said.
"Our plan was to see if we could take some time on our opponents today. That happened and I'm just so proud to do it today and do it for my daughter."
A smiling Kuss held onto the leader's red jersey with a brave ride to finish runner-up ahead of Roglic in a Jumbo-Visma 1-2-3.
"I've done the Tourmalet a few times, and it's quite long, but to be honest, it's a bit overhyped," Kuss had said on Thursday, and the American looked comfortable throughout.
On the second climb to Col d'Aubisque, both Evenepoel and Joao Almeida were dropped.
By the time they hit the next ascent, Evenepoel was over two minutes adrift of the leaders, and it only got worse from there.
Any early attacks had all been reeled in by a Jumbo-Visma team looking to control the race.
The Jumbo-Visma trio of Kuss, Roglic and Vingegaard were part of a breakaway group with 60 kilometres left, along with Marc Soler, who began the day in second place, and his UAE Team Emirates teammate Juan Ayuso.
TACKLING THE TOURMALET
As they hit final descent before tackling the Tourmalet, Almeida was over two minutes down while the gap to Evenepoel had drifted out to over seven minutes.
At the bottom of the Tourmalet, with 19 kilometres still to race, it was clear the stage winner would come from a 22-man bunch. Jumbo-Visma's Robert Gesink and Wilco Kelderman led the way, with the team's main men biding their time.
With Gesink's work done, he too dropped away from the leaders, and the group was whittled down to 14 riders heading into the final 12 kilometres.
Vingegaard made a couple of attempts to get away, and when he did nobody was going to catch the twice Tour de France winner.
Enric Mas tried to get away from the chasers but Kuss was alert to the danger. With the five-man group left to battle behind Vingegaard, it was Kuss who attacked in the final kilometre to close the gap.
On the line Kuss came in 30 seconds behind his team mate, with Roglic just three seconds further back on a day when Jumbo-Visma took control of la Vuelta.
At the start of the stage, Kuss had a 26-second lead over Marc Soler but Roglic is now in second place, one minute 37 seconds behind with Vingegaard just seven seconds further adrift in third place.
After his win, Vingegaard, who now has the best climber's polka dot jersey, was asked whether the 1-2-3 finish had been Jumbo-Visma's plan for the stage.
"That's even better than the plan," the Dane replied.
For Evenepoel, this year's Vuelta challenge is already over. The Belgian trailed in over 27 minutes behind the stage winner.
Saturday's stage 14 is another mountain stage, a 156.5 kilometre ride from Sauveterre-de-Bearn to Larra-Belagua.