Adriano's stunning first-half effort ensured Barcelona made it two wins from two Primera Division games under new boss Gerardo Martino.
Barca coasted to a 7-0 win in their opener aganist Levante last weekend, but they were made to sweat at La Rosaleda.
The Brazilian grabbed victory with a memorable 25-yard strike just before half-time although the hosts almost equalised after the break when Fabrice shot against the post from close range.
Atletico Madrid made it two wins from two for the new Primera Division season by ruthlessly dispatching local rivals Rayo Vallecano.
Diego Simeone's side, who opened their campaign with a 3-1 win at Sevilla, were three goals to the good by half-time as Raul Garcia, Diego Costa and Arda Turan all found the back of the net.
Rayo decided to tighten up in defence after Tiago added a fourth following the break but Garcia made sure there was no escaping a rout with one last goal on the stroke of full-time.
Levante bounced backed from their opening-weekend mauling at Barcelona with a home draw against Sevilla.
Joaquin Caparros' side may still be awaiting their first goal of the new Primera Division campaign but they at least banked their first point in front of their home fans - following a 7-0 humiliation at Camp Nou.
Sevilla too earned their first point of the season after they were beaten 3-1 at home to Atletico Madrid last weekend.