The two launched attacks immediately after the start of the 118.5-kilometre stage to the Aramón Formigal ski resort. Froome quickly found himself on his own as a 14-man group with Quintana and Tinkoff's Contador – both with two teammates – rode away.
"It was a dia grandioso," Quintana said with the leader's jersey, now with a healthy 3-37 lead over Froome in second place.
"We were very attentive at the start of the stage because we knew it was complicated. Contador played a key role so that this day could be so important."
Quintana pushed the difference on Froome from 54 seconds. He placed second on the stage behind winner Gianluca Brambilla (Etixx–Quick Step) and gained 2-43 on Froome.
The gains made came thanks to the alliance with Contador and his teammates, who helped push over two passes to reach the base of the 14.5-kilometre final climb. Once on the climb, Quintana began to ride at the front of the group while behind Froome had to pull after teams Astana and Orica-BikeExchange ran out of gas.
"I wasn't so optimistic at the start because the climbs didn't seem that difficult, but I attacked like crazy at the start," Contador explained. "I didn't even know who was on my wheel and I knew that it was a dangerous bet, but look at what happened.
"The nice thing is we re-wrote the script. We don't have a big strong team like Movistar or Sky, so we had to look for allies."
Contador leapfrogged Simon Yates, who was in the Froome group with Orica teammate Esteban Chaves, and moved from sixth to fourth overall. He now sits just five seconds behind Chaves in third and 25 behind Froome in second.
On Monday, the stage will climb midway in and end with a flat run in Peñiscola. After the rest day on Tuesday, the race will climb up the Alto Mas la Costa for its finish and on Wednesday, it will time trial through Calpe.
"I hope to have the red jersey now all the way to Madrid," added Quintana. "The gap now to Froome is a big difference."
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