The Catalan club opened their campaign with a 7-0 triumph against a hapless Levante side last week, but were short of their best form as well as injured star man Lionel Messi at La Rosaleda.
Adriano 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 joined rivals Barcelona on six points by ruthlessly dispatching local rivals Rayo Vallecano 5-0 at the Vicente Calderon.
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.
Second-half goals from Charles and Nolito ensured 10-man Celta Vigo claimed their first win of the Primera Division season with battling 2-1 victory at Real Betis.
However, after Borja Oubina was needlessly sent off for collecting a second bookable offence, Ruben Castro got Betis back into the match in the final minute only for fellow forward Chuli to hammer the bar from close range in added-on time as Celta Vigo just about held on.
Levante bounced backed from their humiliation at Barcelona with a goalless draw against Sevilla.