The Slovenian jumped away from a group of top guns 1.5km from the top of the 13.3-km ascent at 7.7% with only Vingegaard able to follow but the Dane lost his rival's wheel some 100 metres further up.
Pogacar rolled over the line eight minutes and 19 seconds behind stage nine winner Michael Woods of Canada, and eight seconds ahead of Vingegaard, who was in full damage control mode in the final kilometre with the Auvergne volcano craters in the background.
"It would be nicer to have stayed with him and have lost no time but as I've said before the first week didn't suit me," said Vingegaard.
"In my opinion there are stages that suit me more, so being in yellow after that first block of racing is good for me. I'm looking forwards to the Alps."
Overall, Vingegaard, who had struck in the Col de Marie Blanque in the first Pyrenean stage last week before losing some ground a day later in the ascent to Cauterets-Cambasque, leads Pogacar by 17 seconds.
The deafening cheers of the crowd made way to an eerie near silence as fans and team cars were not allowed in the final four kilometres of the climb.
Pogacar and Vingegaard kept an eye on each other, jerseys zipped up in the haze of the Giant of Auvergne before the 2020 and 2021 champion attacked, in a move reminiscent of Raymond Poulidor's against great rival Jacques Anquetil on the same slopes back in 1964.
That day, Poulidor, who died in 2019 and received a huge amount of love and tributes as the stage started from his home town, gained 42 seconds to narrow the gap in the general classification to 14.
He eventually lost the Tour to Anquetil for 55 seconds.
There were two races on Sunday as a 14-man breakaway group opened a big gap and Matteo Jorgenson, who had gone solo with 46km left, tackled the punishing climb with a 15-minute lead over the main bunch.
"It was so loud, so many fans, it was deafening and then when you pass the 4-km mark it is total silence," said Woods.
Among the podium contenders, Australian Jai Hindley retained his third place overall after crossing the line more than a minute behind Pogacar.
Fourth-placed Carlos Rodriguez of Spain kept Ineos-Grenadiers' podium chances alive despite losing 60 seconds to Pogacar.
He now trails 2022 Giro d'Italia winner Hindley by 1:42.
Three Britons, Adam Yates and his brother Simon, and Tom Pidcock are fifth, sixth and seventh overall, respectively.
French hopes of a podium finish on the Champs Elysees took a major blow as David Gaudu was again unable to follow the big guns and dropped to eighth overall, 6:01 off the pace after being dragged to the line by Groupama-FDJ team mate Thibaut Pinot, a former podium finisher who is on his last Tour.
Monday is a rest day before the peloton returns to the mountains on July 14 with the lung-busting ascent to the Grand Colombier (17.4% at 7.1%) - the first of five mountain treks before Paris.