Robert Lewandowski scored twice as Bayern Munich opened Group E with an impressive 3-0 win in Barcelona.
Thomas Muller's deflected effort put the Germans in front in the 34th minute. Lewandowski then scored for the 18th club game in a row in the 56th minute after the ball came back off the post.
The Poland striker grabbed his second five minutes from time when he knocked the ball in after Serge Gnabry's shot had also hit the upright.
Juventus claimed their first win of the post-Cristiano Ronaldo era by beating Malmo 3-0 with a flurry of first-half goals.
Juventus are yet to win in Serie A this season, the last two games being losses after the departure of Ronaldo to Manchester United.
But Alex Sandro's 23rd-minute opener was followed by two goals in the space of 60 seconds. Paulo Dybala scored from the penalty spot before and Alvaro Morata added the third on the stroke of half-time.
Juventus will be hoping they can take some of the momentum into Sunday's Serie A clash against AC Milan.
Cristiano Ronaldo scored again for Man Utd, but they were beaten 2-1 late on by Young Boys in their Champions League opener after Aaron Wan-Bissaka's red card.
Romelu Lukaku's second-half header saved an under-par Chelsea as they began their Champions League defence with a 1-0 win over Zenit at Stamford Bridge.
Sevilla were held to a 1-1 draw at home to RB Salzburg after a chaotic first half in which four penalties were awarded, three to the visitors, while the hosts were forced to play most of the second half with 10 men.
Karim Adeyemi missed from the spot early one, but the Austrian's earned a second penalty in the 21st minute, which Luka Sucic managed to tuck away.
Salzburg then won another penalty just before the break, but Sucic hit the post with his second effort. Sevilla were then awarded a penalty of their own to mark the first time in Champions League history in which four penalties were awarded in the first half.
Ivan Rakitic confidently dispatched his kick to pull Julen Lopetegui's side level, but the hosts were soon on the back foot again early in the second half when striker Youssef En-Nesyri was sent off for a second booking.
Sevilla 'keeper Bono made a string of saves as the Spaniards held on for a draw, but Lopetegui's 10 men could have won the match with a late header from substitute Erik Lamela which went narrowly wide.
In Tuesday night's other Group F game, Villarreal and Atalanta drew 2-2 at El Madrigal.
Remo Freuler gave the Italians the lead in the sixth minute, but Manuel Trigueros hauled Villarreal level shortly before half-time.
Villarreal - who beat United on penalties to lift the Europa League trophy - equalised through Arnaut Danjuma in the 73rd minute. Robin Gosens, though, struck late to earn Atalanta a point.
Dynamo Kiev saw a stoppage-time goal from Mykola Shaparenko ruled out following a VAR review as they had to settle for a point following a 0-0 draw at home against Benfica.
It finished goalless between Lille and Wolfsburg - but there was late drama after a penalty for the French side deep into stoppage-time was overturned by VAR.
Lille had earlier seen a goal for Jonathan David ruled out by VAR at the start of the second half before Wolfsburg had defender John Brooks sent off for two cautions.