In a gripping finale, Quintana powered clear of Froome and sliced almost a minute and a half off the Briton's lead, but he could not do quite enough to prevent the Team Sky rider from almost certainly celebrating victory on Sunday.
With only the procession into Paris remaining, Froome leads Quintana by 1:12 after the Movistar rider attacked repeatedly and pushed the 30-year-old to the limit in a punishing ascent.
The day's laurels went to Frenchman Thibaut Pinot, who salvaged an under-par Tour with victory on the race's 'Queen stage'.
Spain's Alejandro Valverde is third, 5:25 off the pace.
Quintana was second in the 110-km stage from Modane-Valfrejus, 18 seconds behind Pinot, with Canadian Ryder Hesjedal taking third place 41 seconds adrift.
Froome, who could not respond when Quintana attacked less than four kilometres into the 13.8-km ascent, ended the stage in fifth place, 1:38 off the pace after being helped to the summit by team mate Richie Porte.
"It was really difficult today but my team mates were fantastic, especially Richie Porte. What can I say, it's a dream," Froome told reporters.
Quintana rued losing lost 1:30 on the second stage after finding himself on the wrong side of a bunch split.
"It was our last chance today, we tried in the col de la Croix de Fer but failed, so we tried again in the final climb," he said. "I think I lost the Tour during the first week."
Froome, who claimed his first Tour title in 2013, was once again booed by the crowd on the final climb.
He will become the first rider since Carlos Sastre in 2008 to win both the yellow and polka dot jerseys, although the Spaniard only claimed the mountain classification after Austrian Bernhard Kohl was stripped of the title following a retroactive dope test.
The green jersey for the points classification will go to Slovakian Peter Sagan and Quintana will don the white jersey for the best Under-25 rider.
Froome, who lost 30 seconds to Quintana on Friday, spent another tough day in the saddle.
Frenchman Alexandre Geniez attacked from the start and broke away with three others but the FDJ rider went out on his own in the climb to the Col de la Croix de Fer.
Among the group of favourites, Valverde attacked about five kilometres from the top of the col de la Croix de Fer and was quickly followed by Quintana.
Helped by Porte, Froome managed to catch Quintana at the top and the group stayed together in the descent.
Pinot and Hesjedal then attacked and the Frenchman, who got help on the way from his FDJ team mate Geniez, powered away six kilometres from the finish, riding his way through hundreds of flare-waving spectators.
The 25-year-old, third overall last year, snatched his second Tour stage win after claiming his first on his debut in 2012.
(Editing by Toby Davis and Ed Osmond)
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