Borussia Dortmund and Inter Milan also have perfect records in Germany and Italy, while Manchester United’s new signing Anthony Martial kept up his impressive start in England.
Here is a look at Europe’s major leagues on Sunday:
Messi scored a pair of goals and Neymar added another as Barcelona Easily beat Levante 4-1 in the Spanish league.
Marc Bartra also netted for the defending champions, which kept a two-point lead over Real Madrid with their fourth consecutive win to start the season.
Villarreal and Celta Vigo also won Sunday to reach 10 points alongside Madrid.
Barcelona could have won by an even bigger margin had Messi not had his second penalty miss of the season in the second half.
Messi’s first goal came from a penalty in the 61st after Neymar was fouled inside the area, but the Argentine playmaker sent his second spot kick over the crossbar in the 76th after he had been pulled down by a defender.
Villarreal moved to third place by beating Athletic Bilbao 3-1, while Celta defeated winless Sevilla 2-1.
Martial continued to repay the hefty fee Manchester United paid Monaco for him by scoring twice to spearhead a 3-2 win over Southampton in the Premier League. Martial became the most expensive teenager ever in a deal that raised eyebrows on deadline day, but now has three goals in two league games.
Danny Ings scored his first goal for Liverpool but couldn’t prevent his team from extending its winless streak in the league to four games after a 1-1 draw against Norwich at Anfield — despite Daniel Sturridge returning after a lengthy injury layoff. In the day’s early game, Son Heung-min netted his first league goal to give Tottenham a 1-0 win over Crystal Palace.
Inter Milan remained the only perfect team in Serie A with a 1-0 win at Chievo Verona, bolstering the status of Roberto Mancini’s squad as a possible title contender.
Also, four-time defending champion Juventus beat 10-man Genoa 2-0 for its first victory with Paul Pogba involved in both goals.
Roma’s 38-year-old captain Francesco Totti scored his 300th goal for Roma but the Giallorossi were held to a 2-2 draw at home by Sassuolo.
In Verona, Inter’s Mauro Icardi scored shortly before halftime, slotting in from the edge of the area after taking a pass from offseason signing Geoffrey Kondogbia.
Through four rounds, Inter is two points ahead of Torino, which got two goals from former Italy international Fabio Quagliarella in a 2-0 win over Sampdoria.
Fiorentina is third, three points behind, after beating Carpi 1-0 with a first-half goal from Khouma Babacar.
Japan midfielder Shinji Kagawa scored one goal and set up another as Borussia Dortmund beat Bayer Leverkusen 3-0 to reclaim top spot in the German Bundesliga.
Dortmund moved back ahead of Bayern Munich on goal difference with both sides perfect after five games played. Bayern won 3-0 at promoted Darmstadt on Saturday.
The win over Leverkusen was Dortmund’s club record 11th win across all competitions since the season began under new coach Thomas Tuchel.
Elsewhere, first-half goals from Alexander Esswein and Paul Verhaegh three minutes later were enough for Augsburg to defeat visiting Hannover 2-0; and Leroy Sane’s second-half strike was enough for Schalke to win 1-0 at Stuttgart to climb into fourth.
Goals from Jonathan Bamba and Robert Beric lifted Saint-Etienne to a 2-0 win over Nantes in the French league, a fourth consecutive victory that lifted the club one point behind leader Paris Saint-Germain.
Bamba, a highly-praised winger who has been nurtured by Saint-Etienne, put the hosts ahead in the 25th minute with a shot in the top corner after beating two Nantes players. Slovenian striker Beric doubled the hosts’ lead in the 48th with a clinical finish after receiving the ball with his back to the goal.
Saint-Etienne’s fierce rival Lyon was held to a 1-1 draw at 10-man Marseille in an ill-tempered match that was interrupted for 20 minutes because of crowd trouble.
Monaco stumbled to its second home loss this season, losing 3-2 against Lorient.