Colombian Quintana launched a late attack on the final climb of the 188-km stage from Lugones to Lagos de Covadonga to pass Robert Gesink and held his own for the remaining 2.4 km of the stage, finishing 24 seconds ahead of the Dutchman.
Tour de France winner Froome produced a strong display on the second half of the final climb to overtake fellow favourites Alberto Contador and Alejandro Valverde and end the race third, one second behind Gesink.
Tuesday is the first rest day of the tour, which ends on Sept. 11 in Madrid.
The Tour recommences on Wednesday with stage 11, a 168.6-km ride from Colunga to Pena Cabarga.
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Quintana leads nearest challenger and Movistar team mate Valverde by 57 seconds and Froome by 58, while Contador dropped two minutes 54 seconds behind the leader in fifth after finishing the stage in eighth place.
"It fills me with pride to leave my name as a winner on this mountain and in the way we did it. I had always dreamt of winning here," Quintana told reporters.
Despite gaining 25 seconds on Froome, Quintana said the Briton was still the favourite.
"He's still very close and we're going to try and keep doing what we've done now to get as far away from him as possible," added Quintana.
Three-time winner Contador admitted this year's Tour of Spain was slipping from his grasp.
"The Tour is looking like an uphill struggle for me," said the Spaniard.
"The ascent proved really tough for me and it was a bad day because of the time I lost. Now I need to recover and take advantage of the rest day as much as possible because Wednesday will be another battle."
(Editing by Neil Robinson and Clare Fallon)
