Quintana's compatriot Johan Esteban Chaves climbed up to third overall after finishing ninth in the 196km journey from Urdax-Dantxarinea to Col d'Aubisque.
Froome, bidding to become the first rider in 38 years to win the Tour de France and the Vuelta in the same season, fended off several attacks from Movistar rider Quintana but was unable to make up any ground on the leader.
Gesink, who missed the Tour de France through injury after crashing at the Tour of Switzerland, attacked during the punishing high-mountain 21km finish having been part of a 40-man breakaway that pulled clear earlier.
Frenchman Kenny Elissonde and Russia's Egor Silin were second and third respectively, seven and nine seconds behind Gesink.
"I've not had a great season but I'm happy to be back," Lotto NL-Jumbo rider Gesink told reporters. "I had to keep fighting to the last...ultimately it was enough for the victory."
Sunday's medium-mountain Stage 15 is a 118.5km ride from Sabinanigo to Sallent de Gallego. The race ends in Madrid on Sept. 11.
(Reporting by Ed Dove; Editing by Tony Jimenez)
