France's Pierre Latour crossed 28 seconds behind Woods with Slovenian Matej Mohoric a further seven seconds adrift as the attention switched to the key fight between the favourites several minutes down the steep slopes.
Two-time Tour winner Tadej Pogacar launched an attack from 600m on the baking slopes but defending champion Vingegaard held his cool and followed the Slovenian over the summit finish line just seconds behind.
Two fancied British riders -- former Vuelta champion Simon Yates and up-and-coming Ineos star Tom Pidcock -- were a further 43 sec behind Vingegaard.
Vingegaard leads the race ahead of Monday's rest day by 17 seconds, as the expected fireworks never materialised on a hot July day.
Woods was surprised with his win after managing to get into the day's escape which led by 16 minutes on the tough slopes where no fans were allowed.
"Because there were no fans at the end it was almost like Covid period climbing," said Woods.
"Up to that point it has been so loud. But it was so silent up there," he said after he overtook American Matteo Jorgenson in the home straight.
Jorgenson of Movistar had been stung on the way to the dormant volcano and sought medical assistance after a wasp got inside his helmet.
The world heritage site in central France provides quite a spectacle and experience in itself.
"You can see the volcano from way away, that was really cool," said Woods.
"This was my career goal and today was one of the hardest races you could possibly do."
POGACAR GAMBLES AGAIN
Pogacar was a full 8min 19sec slower than Woods after the 182km stage nine as he finished first from the main contenders after his trademark late burst.
Vingegaard revealed his team had held back.
"We didn't want to go for the stage here, we didn't want to pull because in my opinion this stage didn't suit me," explained the Dane.
"I'm looking forward to getting to the Alps," he continued. "There are stages that suit me a lot better than this."
The former fish factory worker, who won last year and was second behind Pogacar in 2021, said he tries only to focus on himself.
"I don't really think about him," he said of Pogacar.
"I just think about maintaining my best shape, but for the plans, he's the one you have to think about," he conceded.
It has been 35 years since the Tour last raced the slopes of the giant volcano where former stars Raymond Poulidor and Jacques Anquetil once competed shoulder-to-shoulder.
On Sunday, Poulidor's grandson the Dutch cycling star Mathieu van der Poel was reduced to tears when presented with one of his old bikes.
He also raced the stage on a special edition Poulidor bike with old black and white images of France's all-time favourite cyclist, and was cheered all the way.