Romain Bardet (AG2R La Mondiale) crossed the line alone in second place, 1:35 back, with Alexey Lutsenko (Astana) completing the podium.
“It was a really grand day for me, and it was within the team strategy. I was ready for it,” said Movistar’s Quintana.
“It’s a beautiful win, and on a stage like this, with this type of profile, it makes me emotional.”
On the final Galibier climb a group of five, which included Bardet, Quintana, Lutsenko, Damiano Caruso (Bahrain-Merida) and Michael Woods (EF Education First) led the stage.
Quintana counter-attacked a move by Woods and became the solo leader with 26km to go.
Bardet and Lutsenko dropped the rest of the group and started chasing Quintana when he was 6km from the top of the climb.
The leader had 50 seconds on the chasers 3km from the summit, with 22km left.
Bardet dropped Lutsenko as Quintana reached the summit alone and began the descent to Valloire and ultimately to the stage victory.
Julian Alaphilippe (Deceuninck-Quick-Step) retained the yellow jersey, while Egan Bernal (Ineos) moved from fifth to second overall, with his teammate and defending champion Geraint Thomas dropping to third.
“We wanted it to be hard but the pace wasn’t there,” said Thomas.
“The call was made for Egan to go and hopefully that would kick if off a bit, but it didn’t. That’s when I went as well, just to test.
“But at least Egan gained some time on everyone else.”
Peter Sagan (Bora-Hansgrohe) held on to green jersey, with Bardet taking the polka dot jersey from Tim Wellens (Lotto Soudal). Bernal remained in white.