LaLiga champions Barcelona claimed an impressive 1-0 win away to Athletic Bilbao on Sunday night thanks to an Ivan Rakitic goal after 20 minutes.
The Basque side had plenty of chances in an open game but Barca displayed the greater quality in front of goal and after Luis Suarez had released Arda Turan down the left, the Turkey international cut inside onto his right foot and his cross found Rakitic racing into the box to head home from six yards.
Suarez almost added an injury-time second as he had an effort cleared off the line, but Luis Enrique's men join bitter rivals Real Madrid, who beat Celta Vigo on Saturday, on six points from two games.
Newly-promoted Las Palmas have also taken a maximum six points after hammering Granada 5-1 at the Estadio de Gran Canaria.
The Canary Islanders, who were surprise 4-2 victors away at Valencia last Monday, again looked potent up front and took a 23rd-minute lead through Nabil El Zhar.
French forward Jeremie Boga equalised for the visitors on the stroke of half-time but Las Palmas were back in front after five minutes of the second period through Kevin-Prince Boateng.
El Zhar added his second after 69 minutes before substitute Momo provided a fourth three minutes later. Argentinian striker Sergio Araujo rounded off the scoring with five minutes remaining.
Another promoted side, Alaves, have made a decent start too, if not as spectacular as Las Palmas.
They followed up their hard-fought 1-1 draw away to Atletico Madrid last weekend by taking a point at home to Sporting Gijon on Sunday.
Villarreal and Sevilla's match at El Madrigal finished goalless as the visitors' goalkeeper Sergio Rico performed heroics to keep Fran Escriba's men at bay.
The Yellow Submarine showed no signs of a hangover from their Champions League exit to Monaco in midweek but were unable to claim all three points as Rico denied them time and again, including tipping a low Manu Trigueros shot onto the crossbar.
Jorge Sampaoli's Sevilla - who won their opening game of the season 6-4 against Espanyol - almost snatched the points in the final seconds when Pablo Sarabia struck the post direct from a corner.