The Movistar team leader finished the stage 30 seconds in front of Froome (Sky). With a record breaking climb up the mountain, he reduced his general classification deficit to the Briton to 2min 38sec ahead of tonight's final showdown on Alpe d'Huez.
Quintant's Movistar team-mate Alejandro Valverde kept his third place in the overall standings after Stage 19. With one mountain stage to go, Valverde will have to defend this position with all the energy he has left after he lost time, from the final meters of the Col du Glandon, against a new big threat: Vincenzo Nibali (Astana).
Nibali controversially attacked while Froome was pulled over on the side of the road removing a pebble from his wheel and went on to win the 138km stage in a time of 4hr 22min 53sec.
When ‘The Shark' attacked, José Herrada (Movistar) lead the charge from the chase group. He limited losses to the Italian, which grew to 2min 10sec on the foot of La Toussuire, the decisive and final ascent of the stage.
With 6km remaining, Nairo Quintana launched a series of strong accelerations. As Froome dropped back Quintana’s gap increased to 30 seconds into the fastest part of the climb and the plain towards the finish.
Fuelled by the knowledge that Froome was behind him, fans of the race pointed out on Twitter that Quintana set a new record for the La Toussuire ascent.
“Overall, the day went pretty well, but we missed that final touch,” said Quintana after the stage. “We initially tried to control Nibali, thinking that Sky would start chasing later on, but it's true that a rider like Vincenzo was really far in the overall standings in that moment, and they didn't have to care about him.
“We kept pushing with José Herrada, who was fantastic, and Tinkoff(-Saxo) pulled a bit later on to gain some time back. I jumped when I thought it was the right moment and place to put a good distance on Froome, but it wasn't enough compared to what we thought."
Behind, Valverde fought bravely and crossed the line, 2min 26sec behind Nibali, in sixth place. He maintained his third overall at 2min 25sec with a 1min 19sec margin over Nibali who moved from seventh to fourth overall.
With time bonuses taken into account, Quintana gained 32 seconds to Froome in the general classification.
Tonight’s 110km final mountain stage includes another ascent of the Hors Category Col de la Croix de Fer before the most famous climb in professional road cycling, Alpe d’Huez.
“There's only one day left, the final mountains for all of us,” said Quintana, who holds the white jersey for the best young rider.
“We'll try to attack stronger and also fight for the stage win. Regarding Alejandro, I think there shouldn't be any problems when riding face-to-face tomorrow; we conceded a big gap today, but I think they'll be together tomorrow.”
While Valverde will fight for third, all eyes will be on Froome and Quintana as they duke it out one last time for the top step in Paris.