The England international scored in both halves as Madrid came from behind to win at Almeria.
Sergio Arribas had the visitors in front in the third minute of Saturday night’s game, but England midfielder Bellingham – who scored the opening-day winner at Athletic Bilbao – equalized 16 minutes later.
After Tony Kroos saw an effort ruled out for VAR at the end of the first half, the former Birmingham Bellingham midfielder put Real ahead on the hour and Vinicius Junior made it 3-1 with 17 minutes remaining.
Athletic Bilbao ran out 2-0 winners at Osasuna, with goals inside the first 20 minutes from Inaki Williams and Gorku Guruzeta doing the damage. The visitors had substitute Chimy Avila late on.
Real Sociedad and Celta Vigo drew 1-1, with Oscar Mingueza earning the visitors a last-minute point after Ander Barrenetxea had opened the scoring for Sociedad midway through the first half.
In Italy, champion Napoli started with a 3:1 win at Frosinone, Viktor Osimhen scored twice.
Abdou Harroui’s seventh-minute penalty put Frosinone ahead again, but Matteo Politano and Osimhen turned the tables on the stroke of half-time before Osimhen sealed the points with 11 minutes remaining.
Lautaro Martinez netted in each half as Inter Milan beat Monza 2-0 at the San Siro, while substitute Federico Bonazzoli got the only goal 15 minutes from time as Verona – who survived last season with a play-off win over Spezia – won at Empoli.
Fiorentina won 4-1 at newly promoted Genoa, with Cristiano Biraghi, Giacomo Bonaventura, Nicolas Gonzalez and Rolando Mandragora securing the points before Davide Biraschi replied for the hosts.
A late penalty from Zakaria Aboukhlal earned Toulouse a 1-1 draw with Ligue 1 champions Paris St Germain.
Kylian Mbappe, who was left out of the squad for the opening weekend’s goalless draw with Lorient after a dispute over his future left him training apart from the first team, came off the bench to slot home a penalty just after the hour mark.
However, Aboukhlal’s free-kick marked a second consecutive draw for the main club.
Lyon is strengthening the table after a 4:1 home loss against Montpellier. Mousa Al-Taamari (2), Arnaud Nordin and Akor Adams were on target for the visitors, with Alexandre Lacazette red-carded for Laurent Blanc’s struggling visitors, who scored their only goal of the game as they went 3-0 up.
In the opening round of the Bundesliga, Donyell Malen scored in the 88th minute of Borussia Dortmund’s 1-0 win over Cologne.
Augsburg and Borussia Monchengladbach shared eight goals in a 4-4 draw, with Bayer Leverkusen, Freiburg, Wolfsburg and Stuttgart also winning.