Champions Juventus started the Italian Serie A season with a victory but Bayern Munich stumbled in the German Bundesliga to a 1-1 draw at Schalke 04.
Zlatan Ibrahimovic scored a hat-trick as French champions Paris St Germain thrashed St Etienne 5-0.
Chelsea underlined their Premier League title credentials with a thrilling 6-3 win at Everton on Saturday to maintain their perfect start to the season.
Spain striker Diego Costa scored twice for Chelsea who top the table with nine points from three games, level with Swansea City who beat West Bromwich Albion 3-0.
Liverpool, with Mario Balotelli making his debut, eased to a 3-0 win at Tottenham Hotspur on Sunday and Aston Villa beat Hull City 2-1 to climb to third.
Champions Manchester City suffered a shock 1-0 home loss to Stoke City and Angel di Maria's debut failed to halt Manchester United's miserable start as they drew 0-0 at promoted Burnley.
Two goals from David Zurutuza helped Real Sociedad seal a remarkable comeback and beat European champions Real Madrid 4-2 on Sunday after going 2-0 down.
Sergio Ramos headed in a corner after five minutes and Gareth Bale scored the second in the 11th, but the home side battled back as the first half progressed and were level by halftime.
Inigo Martinez swept home a cross at the far post after 35 minutes and Zurutuza was left unmarked to head the second. After the break Zurutuza and substitute Carlos Vela took further advantage of defensive lapses to seal a convincing win.
Barcelona continued Luis Enrique's winning start as coach with a 1-0 victory over Villarreal, although Lionel Messi picked up a hamstring strain.
Champions Juventus began the new Serie A season with a 1-0 win at Chievo on Saturday as Massimiliano Allegri made his debut as coach.
AS Roma, second last season, beat Fiorentina 2-0 and Napoli snatched a 2-1 win at Genoa with a stoppage-time goal by Jonathan de Guzman.
AC Milan got off to a winning start under new coach Filippo Inzaghi by beating Lazio 3-1 at San Siro.
Inter Milan goalkeeper Samir Handanovic saved a first-half penalty to earn them a goalless draw at Torino, although defender Nemanja Vidic was sent off on his debut for Walter Mazzarri's side.
Champions Bayern Munich, with new signing Xabi Alonso in the starting lineup, stumbled to a 1-1 draw against gutsy Schalke 04 on Saturday.
Robert Lewandowski scored his first competitive goal for the Bavarians in a one-sided first half in which the visitors should have added to their goal tally.
Schalke roared back in the second half to level as Benedikt Hoewedes scored to secure their first point of the season.
Bayer Leverkusen remained top with their second win from two games, a 4-2 victory over Hertha Berlin, while last-season's runners-up Borussia Dortmund edged past Augsburg 3-2 to bounce back from their opening defeat.
Paris St Germain hit their stride in Ligue 1 as Zlatan Ibrahimovic scored a hat trick in their 5-0 demolition of St Etienne at the Parc des Princes on Sunday.
PSG were helped by St Etienne keeper Stephane Ruffier's early blunder before Ibrahimovic took his season's tally to five goals.
PSG are second with eight points from four games, two points behind Girondins de Bordeaux who dropped their first points in a 1-1 draw against Bastia earlier on Sunday.
Olympique de Marseille secured a second consecutive win by beating Nice 4-0 on Friday while Monaco's mediocre start continued with a 1-1 draw against Lille that left them 15th and with Colombia striker Radamel Falcao not on the team sheet.
Second-half goals from Luuk de Jong and Adam Maher maintained PSV Eindhoven's winning start to the season on Sunday as champions Ajax Amsterdam lost for a second successive week.
PSV beat Vitesse Arnhem 2-0 at home for a fourth successive win and a three-point lead over PEC Zwolle, whose 100 percent record ended at NAC Breda, and Groningen.
Ajax, seeking an unprecedented fifth successive title, were beaten 2-0 at Groningen a week after they had lost 3-1 at home to PSV.
Champions Benfica drew 1-1 with Sporting Lisbon in their derby at the Stadium of Light on Sunday.
Argentine Nicolas Gaitan gave the Eagles a 12th minute lead and Algeria forward Islam Slimani equalised eight minutes later for Sporting as Benfica dropped their first points of the season.
Porto and Vitoria Guimarares are the only teams with maximum points from three games. Colombia forward Jackson Martinez scored twice in the last 10 minutes in Porto's 3-0 win over Moreirense and Vitoria won by the same score at Belenenses.
Braga and Benfica are next in the table with seven apiece.