Froome and Quintana finished 1min 47sec further back as the Brit neutralised multiple attempts by the Colombian to extend his 54sec lead in the general classification.
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The Colombian dealt a body blow to the Tour de France victor when he joined a breakaway with Alberto Contador in the early kilometres of stage 15.
Esteban Chaves of Colombia remains third overall, 20 seconds adrift of Froome.
An early break on Sunday’s 118.5-kilometre stage from Sabinanigo to Aramon Formigal caught Froome out, though, and Quintana took advantage to seize control of the GC race.
“Really I’ve run out of strength”.
He said: “My team helped me out so much. This morning I felt flawless”.
The Dutch rider surged ahead of two rivals to win the 196km (122-mile) route for Lotto NL-Jumbo, finishing at the Col d’Aubisque category-one summit after almost six hours.
“We knew that the breakaway would go to the end because we had a huge advantage”.
Yates made the most significant move of the day as he attacked 40km from the finish with ORICA-BikeExchange using their riders in the breakaway to help the stage six victor open up a sizeable advantage on the red jersey group.
Instead, Froome had to rely on Astana, despite the Kazak team having no rider in general classification contention, to set the pace for most of the stage just to keep the gap from extending beyond three minutes.
Quintana’s Movistar team launched a separate attack soon after to bridge the gap to the leaders, but Froome was left isolated and with most of his team-mates even further behind.
Froome will still have the individual time trial, one of his strengths, on stage 19 to look toward.
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Froome is trying to become the first man in 38 years to win the Tour de France and Vuelta in the same year. This probably sounds the death knell of his ambitions to sign a top-10 in the three major Tours in the same season after finishing third in the Giro in May and 6th in the Tour de France in July. “This may seem a bad day but on the other hand, I see that I have good legs and I’m not so bad finished the stage”, consoled the Spaniard.
