Sanchez produced a solo ride to victory 2-km after completing the notoriously difficult Arrate climb.
Compatriot Alejandro Valverde was second, two seconds back, and regained the overall lead from fellow Spaniard Oscar Freire.
Dutchman Robert Gesink finished third.
One of two teams in cycling's top ProTour league with no wins this season, the other being Omega Pharma-Lotto, Sanchez said his triumph would ease the pressure on Euskaltel-Euskadi.
"Taking the first victory of the year is always important, this time much more so than usual, and winning in Arrate like [Spain's top stage racer] Alberto Contador did last year makes me very proud," the 2008 Olympic road-race champion told reporters.
"I had a rough start to this race, losing nearly two minutes on the first day, but that also meant I had a little bit of room to maneuver. I took full advantage of that today." While Sanchez received a Basque beret for his win in the race's most mountainous stage, Valverde again donned the leader's yellow jersey he had worn for the first two days.
"Today I proved I could climb with the best of them but the race is a long way from being won," said Valverde.
"Gesink and (Chris) Horner could be real threats in these last two days - they showed today they were very strong." Valverde has a one-second advantage over American Horner and Gesink.
"It's not going to be decided until the last meter of the last stage," said Valverde.
The race finishes on Saturday with an individual time trial in Orio.
