Karim Benzema's header gave Real Madrid a 1-0 win at Sevilla on Sunday, allowing Los Blancos to climb to second in La Liga.
Athletic Bilbao earned their third league win from five with a 2-0 victory over Alaves, and the Lions are the new leaders in the Spanish top flight.
Espanyol's poor start to the season continued, with the Catalans defeated 3-1 at home by Real Sociedad.
Elsewhere, Getafe beat Mallorca 4-2, and Valencia drew 1-1 with bottom side Leganes.
Getafe 4-2 Mallorca
Espanyol 1-3 Sociedad
Valencia 1-1 Leganes
Athletic Club 2-0 Alaves
Sevilla 0-1 Real Madrid
La Liga Standings (Played, Won, Goal Difference and Points)
1. Athletic Club: 5, 3, +5, 11
2. Real Madrid: 5, 3, +4, 11
3. Granada: 5, 3, +6, 10
4. Real Sociedad: 5, 3, +3, 10
5. Sevilla: 5, 3, +3, 10
6. Atletico Madrid: 5, 3, +1, 10
7. Villarreal: 5, 2, +4, 8
8. Barcelona: 5, 2, +3, 7
9. Levante: 5, 2, +1, 7
10. Osasuna: 5, 1, +1, 7
11. Getafe: 5, 0, +1, 6
12. Valencia: 6, 1, -2, 6
13. Alaves: 5, 1, -2, 5
14. Real Betis: 5, 1, -3, 5
15. Valladolid: 5, 1, -3, 5
16. Celta Vigo: 5, 1, -3, 5
17. Mallorca: 5, 1, -4, 4
18. Espanyol: 5, 1, -6, 4
19. Eibar: 5, 0, -3, 2
20. Leganes: 6, 0, -6, 1
Benzema's goal was the difference for Real on their travels to Seville, boosting his team's league standing.
The hosts drew a blank as they failed to hit the target during the match, and the visitors snatched the points with the only goal of the game after 64 minutes.
Sevilla had a slight edge in possession, but it was Benzema who headed the winner after combining with Gareth Bale and Dani Carvajal.
The UEFA Champions League's account highlighted the Frenchman's form:
Athletic Club went top of La Liga after they beat Alaves 2-0 at the San Mames Stadium.
The video assistant referee ruled Ruben Duarte had fouled Inaki Williams in the box, and Raul Garcia slotted home the penalty to break the deadlock after 38 minutes.
Victory was assured with 18 minutes remaining after Iker Muniain found the back of the net.
La Real were clinical during their trip to Catalonia, and the visitors gained control early on. Willian Jose opened the scoring after linking with Mikel Oyarzabal for Sociedad on 18 minutes, and the Brazilian striker completed his brace after 34 minutes.
An own goal by Joseba Zaldua gave Espanyol hope with 19 minutes remaining, but the Basque side scored a third through Alexander Isak four minutes later.
Getafe hit four against Mallorca on the outskirts of the capital, gaining a mid-table berth.
Iddrisu Baba's own goal gave the hosts an early advantage after seven minutes, and Jorge Molina scored from the penalty spot when Joan Sastre fouled Jaime Mata in the box on 31 minutes.
Mata fed Allan Nyom for the third after 63 minutes before the visitors hit back with two quick goals. Ante Budimir scored twice in seven minutes for Mallorca, but Angel Rodriguez sealed victory with just six minutes on the clock.
Los Ches failed to kick-start their campaign after Leganes earned a draw at the Mestalla Stadium.
Rodrigo Moreno was dangerous for Valencia in the opening stages, and the forward was fouled by Ruben Perez in the box after 21 minutes, allowing Daniel Parejo to open the scoring from the spot.
However, La Liga's bottom club scored an equaliser when Oscar Rodriguez struck 10 minutes before half-time, with the visitors winning their first point of the season. Valencia have won only once in their first five league games.