Etixx Quick Step rider De la Cruz finished 27 seconds ahead of Belgian Dries Devenyns following a commanding late attack on the final climb of the 164.5km stage from Cistierna to Alto de Naranco and will wear the red jersey in Monday's stage from Lugones to Lagos de Covadonga.
Italian Moreno Moser came third, a further six seconds behind Devenyns.
De La Cruz entered the stage in 15th place but now leads Colombian Quintana by 22 seconds and is a further 19 seconds ahead of compatriot Alejandro Valverde, who moved eight seconds in front of Tour de France winner Chris Froome.
The Briton finished together with Quintana, Spaniard Alberto Contador and Colombian Johan Esteban Chaves, two minutes and 56 seconds behind De La Cruz.
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"I wasn't thinking about the (overall) lead at the end. I was so close to winning the stage, that was my main objective," De La Cruz told Spanish television.
"It was very hard to ride with Devenyns because he was hard to beat, but I am the leader as well, I can't believe it.
"I felt good sometimes but at other times I didn't feel too good and I didn't think I would be good enough to finish it off, but with 15km left I started thinking about reaching the end," added the 27-year-old.
(Reporting by Richard Martin, editing by Ed Osmond)
