Atletico Madrid's Jackson Martinez scored his first goal for the club when he came off the bench and curled in a superb effort to seal a hard-fought 3-0 victory at Sevilla in La Liga on Sunday.
Atletico, the 2014 champions, and Sevilla, last season's Europa League winners, are seen as possible challengers to heavyweights Barcelona and Real Madrid for the title.
By contrast, Valencia, also expecting to be in the mix this season after last year's takeover by Singapore billionaire Peter Lim, could only manage a 1-1 stalemate at home to Deportivo La Coruna, their second draw in two outings.
Spanish and European champions Barca, chasing a sixth La Liga title in eight years, won 1-0 at home to Malaga on Saturday and are one of four teams with a maximum six points, along with Atletico, Celta Vigo and surprise package Eibar.
Celta thumped 10-man Rayo Vallecano 3-0 on Saturday, while Eibar beat more illustrious Basque rivals Athletic Bilbao 2-0 earlier on Sunday.
Eibar were relegated last season but restored to Spain’s top flight after Elche were demoted over unpaid taxes.
Real Madrid got their campaign up and running after a 0-0 draw at promoted Sporting Gijon on the opening weekend when they thrashed last season's second-division champions Real Betis at the Bernabeu on Saturday and have four points in fifth.
At the Sanchez Pizjuan in Seville, Spain midfielder Koke stabbed the ball into the net after tenacious work from Antoine Griezmann in the 35th minute of a scrappy game.
Fernando Llorente, making his debut after joining Sevilla from Juventus this week, came close to an equaliser in the 65th minute but hooked his snap-shot narrowly wide.
Sevilla’s hopes of rescuing a point were effectively snuffed out 12 minutes from time when Gabi’s long-range strike took a big deflection and flew past goalkeeper Beto into the net.
Colombia striker Martinez, who arrived from Porto last month, scored the pick of the goals in the 85th minute before Griezmann struck a similar effort against a post in added time.
"I think we came out to win the match from the start and it was a very complete performance," Gabi told Spanish television.
"Sevilla are our direct rivals and we have taken a big step forward," added the midfielder, whose team hosts Barca on Sept. 12 following the international break.
Koke and Gabi both dedicated their goals to teammate Raul Garcia, who is poised to join Bilbao after agreement in principle was reached on the deal on Sunday.