Chelsea and Barcelona kept on winning in England and Spain while Manchester City and Real Madrid suffered shock defeats.
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