Pogacar, who looked to be struggling in the final ascent of Sunday's 15th stage won by Dutch Wout Poels, attacked just inside the final kilometre on the Bettex climb (7km at 7.7%) but could not drop the Jumbo-Visma rider.
Vingegaard tried to catch Pogacar off guard in the last steep bend but both rolled over the line together, six minutes and four seconds behind Poels, who gave Bahrain Victorious their second stage win this year, a month after the death of team mate Gino Mader at the Tour de Suisse.
On a trek featuring three first-category climbs, Vingegaard and Pogacar put their teams to work but let a large breakaway go after a massive pile-up, signalling they would not go for the stage win and the bonus seconds that go with it, a day after another epic round in their fight for the title.
"I think (in) the time trial there will be a gap, also the next day with one of the hardest climbs in the world (to the Col de la Loze). Also stage 20 could make the difference," said Pogacar.
"Of course, I would have liked to create a gap today. In such a situation with a rider like Jonas it's good to take time when you can but being 10 seconds behind pushes me to be aggressive and that suits me.
"Today is a tie. It was a good day, although three of my teammates crashed," said Vingegaard, whose teammates Sepp Kuss (28), Nathan van Hooydonck (27) and Dylan van Baarle (31) hit the deck with others after one of the Jumbo Visma riders was put off balance by a spectator.
Vingegaard's teammate Wout van Aert (28) thought it was his day to shine when he jumped away from the leading group near the top of the Col des Aravis, followed by Poels and Krists Neilands (28).
The trio quickly caught Marc Soler (29), who had gone solo earlier, only for Latvia's Neilands to crash in the descent as he tried to grab a water bottle from a race motorbike on a left-hand bend.
LOOKING ANXIOUSLY
Poels, Van Aert and Soler opened a 1:30 gap over the breakaway group and when Dutchman Van Aert was expected to be the strongest in the finale, Poels attacked at the foot of the Cote des Amerands (2.7km at 10.9%).
He never looked back and crossed the line 2:08 ahead of Van Aert for his first victory in a grand tour with France's Mathieu Burgaudeau (24) taking third place, three minutes off the pace.
"It's a dream come true. It's my 10th Tour de France, a race I used to watch on TV when I was a kid," said Poels.
Some seven minutes behind, all eyes were on Pogacar and Vingegaard, who was isolated with the 2020 and 2021 winner and the Slovenian's UAE Emirates teammate Adam Yates (30) with just under five km to go.
Pogacar, however, stayed in Yates's wheel with Vingegaard seeming content to follow.
Yates went with 3.5km left, possibly looking to leapfrog Carlos Rodriguez (22) into third place overall.
As Vingegaard and Pogacar slowed down, Spain's Rodriguez came back and made the effort in front with Vingegaard, behind him, looking anxiously over his shoulder as Pogacar lay in wait.
The Slovenian made his move but lacked the stamina to drop Vingegaard, both riders eventually paying for their efforts of the previous days.
Rodriguez is still third in the general classification, 5:21 behind Vingegaard, with Yates in fourth, 19 seconds further back.
