Zlatan Ibrahimovic made a winning return to the AC Milan team as his quickfire double won the Milan derby, while Real Madrid and Barcelona suffered a shock defeats in La Liga.
The 39-year-old striker had missed recent games after contracting coronavirus and only resumed training this week, but scored twice early on.
Ibrahimovic put Milan in front after 13 minutes, tapping the ball in from the rebound after his penalty was saved by Samir Handanovic, and then struck again three minutes later from Rafael Leao's cross.
In a breathless first half, Romelu Lukaku pulled one back for Inter in the 29th minute - the first Serie A goal Milan have conceded this season. Inter were awarded a penalty in the 75th minute but it was revoked for offside in the build-up.
Milan have won all four games so far and top the table with 12 points, while Inter have seven after their first defeat of the season.
Federico Chiesa was sent off on his debut for Juventus as Crotone earned their first point back in Serie A.
Juventus was without Cristiano Ronaldo and Weston McKennie, who both tested positive for COVID-19, and also had several other players out injured.
Simy gave Crotone a surprise lead from the penalty spot in the 12th minute after Leonardo Bonucci fouled Arkadiusz Reca.
Chiesa set up Alvaro Morata for the equaliser nine minutes later but was shown a straight red on the hour following a challenge on Crotone midfielder Luca Cigarini.
Juve thought they had scored a winner shortly after but Morata's effort was ruled out for offside on video review by VAR.
Napoli were 4-1 winners against Atalanta, which was only their second loss since January.
Hirving Lozano scored twice for Napoli and there were also goals for Matteo Politano and Victor Osimhen in the first half. Sam Lammers netted a consolation for Atalanta in the 69th.
The only positive note for Atalanta from a miserable afternoon was the return of key forward Josip Ilicic. He has not played since July after being given compassionate leave because of personal issues.
Striker Fabio Quagliarella set Sampdoria on the way to a comfortable 3-0 win at home to a disappointing Lazio.
The 37-year-old capitalised on poor marking by Lazio to head in from Tommaso Augello's cross in the 32nd minute after Sampdoria had taken control of the game.
Augello added the second himself nine minutes later when a clearance landed at his feet and he thumped a left-foot shot into the net from the edge of the area for his first Serie A goal at the age of 26.
Lazio failed to take their second-half chances and substitute Mikkel Damsgaard wrapped the game up for Sampdoria in the 74th minute after the ball ricocheted around the Lazio area.
La Liga: Real, Barca in shock defeats, Suarez scores for Atleti
Spanish champions Real Madrid suffered a shock 1-0 loss at home to promoted Cadiz - their first La Liga defeat in over seven months.
Honduran forward Anthony Lozano put Cadiz ahead in the 16th minute, darting between Madrid defenders Sergio Ramos and Raphael Varane to meet a cushioned header from Alvaro Negredo and lifting the ball over keeper Thibaut Courtois into the net.
Madrid coach Zinedine Zidane made four substitutions at half-time in a bid to turn the game around and brought on Luka Jovic later on. The Serbian striker soon putt the ball in the net but saw his effort ruled out for a clear offside.
Vinicius Jr missed the target with a free header, while Karim Benzema struck the bar from the edge of the area and Cadiz clung on to claim their first league win over Madrid since 1991.
Barcelona missed the chance to go top of La Liga after slipping to a surprise 1-0 defeat at Getafe.
Jaime Mata scored the only goal from the penalty spot early in the second half and that was enough to lift Getafe up to second, level on points with leaders Real Madrid, who were beaten at home by Cadiz earlier on Saturday.
Lionel Messi hit a post for Barca in the first half, but it was a frustrating evening for the Catalans and Getafe also hit the woodwork in the closing stages.
Barca's first defeat of the season under head coach Ronald Koeman leaves them ninth in the table, three points adrift of Madrid with a game in hand.
Luis Suarez scored an early goal to help Atletico Madrid win 2-0 at Celta Vigo, breaking a short run of 0-0 stalemates against Huesca and Villarreal.
Suarez netted after a pass from Manuel Sanchez in the sixth minute at Celta's Balados Stadium. Yannick Carrasco added a second goal in stoppage time when he headed in a rebound after fellow substitute Joao Flix hit the crossbar with a curling strike.
Celta had their chances to equalise in the first half, only to be stopped by the woodwork and goalkeeper Jan Oblak.
Sevilla will visit Chelsea in the Champions League on Tuesday after seeing their unbeaten La Liga streak ended by Granada.
It was Sevilla's first loss in the league since February, as Yangel Herrera nodded home with eight minutes to play. Herrera has scored five goals in seven matches in all competitions for Granada.
Sevilla played the entire second half with 10 men too with Joan Jordan picking up two bookings for back-to-back fouls in first-half stoppage time.
Bundesliga: Bayern win, Reus and Haaland save Dortmund's blushes
Bayern Munich's Robert Lewandowski and Thomas Mueller scored two goals each as the German champions crushed hosts Arminia Bielefeld 4-1 to move within a point of top spot.
Mueller grabbed an eighth-minute lead with a bit of luck after his first effort was saved but the ball bounced back into his path, while Lewandowski was left with far too much space to score with a well-placed shot in the 26th. With the last kick of the first half, a deflected shot from Lewandowski found the back of the net after good work from Mueller.
Bayern, who had to use changing rooms in a nearby school ahead of the game due to strict Covid-19 regulations, scored again six minutes after the restart when Lewandowski crossed and Mueller volleyed in from close range.
Arminia's Japan international Ritsu Doan was rewarded for a fine performance with a goal in the 58th before Bayern were left with 10 men following the dismissal of Corentin Tolisso for a last-man foul.
Substitutes Marco Reus and Erling Haaland came on to end Borussia Dortmund's barren run against Hoffenheim with a 1-0 victory in the Bundesliga.
Dortmund hadn't won any of the last five meetings between the sides, but Haaland ran on to a ball headed from midfield and kept his cool in front of Hoffenheim's goalkeeper before unselfishly crossing to give Reus a tap-in in the 76th minute.
It had been an uninspiring game from the visitors, the first of seven matches in 21 days, including Tuesday's Champions League opener at Lazio and next week's Ruhr derby against Schalke.
Yussuf Poulsen scored a contender for goal of the season to help RB Leipzig end Augsburg's perfect start with a 2-0 win. Dani Olmo sent the ball on and Poulsen connected with a volley to curl the ball over the goalkeeper and inside the far post.
Elsewhere, Freiburg and Werder Bremen drew 1-1, Hertha Berlin lost 2-0 at home to promoted Stuttgart and Bayer Leverkusen won at Mainz 1-0.
Ligue 1: Marseille back to winning ways
In France, Marseille saw off Bordeaux 3-1 to record a first Ligue 1 win in five matches.
Florian Thauvin fired the hosts into an early lead after only five minutes, with Jordan Amavi also on target in the second half.
An own goal from Pablo wrapped things up ahead of a late consolation from Bordeaux substitute Josh Maja.
Earlier, Lorient came from behind to win 3-1 at struggling Reims, who had midfielder Moreto Cassama and substitute Anastasios Donis sent off during the closing stages.