Liverpool came from behind to defeat Milan 3-2 at Anfield in a meeting of two clubs who have combined for 13 European Cups
Lionel Messi made his first Paris Saint-Germain start in a 1-1 draw at Club Brugge in the Champions League Wednesday, while last season's runners-up Manchester City and Liverpool won high-scoring thrillers.
Real Madrid, the record 13-time European champions, left it late to beat Inter Milan at the San Siro in a section topped by Moldovan debutants Sheriff Tiraspol after the first round of group stage matches.
Argentine star Messi lined up with both Neymar and Kylian Mbappe for the first time, after being limited to just 24 minutes in a PSG shirt since his shock exit from Barcelona.
Ander Herrera, who scored twice in the weekend win over Clermont, gave PSG the lead by steering in Mbappe's cross on 15 minutes, but Brugge soon levelled through Hans Vanaken.
Messi curled against the bar from the corner of the area but PSG then lost Mbappe to injury and had to settle for a point against a team viewed as makeweights in a demanding Group A.
Pep Guardiola's City, beaten 1-0 by Chelsea in May's final in Porto, overcame a hat-trick from Christopher Nkunku to beat RB Leipzig 6-3 at Etihad Stadium.
Nathan Ake headed City ahead from a corner and Nordi Mukiele's own goal doubled their advantage, but Nkunku nodded in to get Leipzig back in the game.
Riyad Mahrez added a third with a penalty before half-time only for Nkunku to head in again early in the second period.
Jack Grealish notched his first Champions League goal following his British record £100m ($136m) signing, with Nkunku completing his treble before Joao Cancelo and Gabriel Jesus strikes wrapped up a wild victory for City.
Leipzig finished with 10 men after former City player Angelino was sent off for a second booking.
Liverpool captain Jordan Henderson hit the winner for Jurgen Klopp's side as they came from behind to defeat Milan 3-2 at Anfield in a meeting of two clubs who have combined for 13 European Cups.
Fikayo Tomori inadvertently deflected in Trent Alexander-Arnold's effort to put Liverpool in front but Mohamed Salah saw his penalty saved by Milan goalkeeper Mike Maignan.
The momentum swung sharply in Milan's favor as Ante Rebic and Brahim Diaz struck twice in as many minutes before the break.
Salah atoned with an equalizer on 49 minutes that was initially discounted for offside before Henderson drove in on the half-volley from the edge of the box.
In the other Group B game, Spanish champions Atletico Madrid drew 0-0 at home to Porto, who were frustrated by a decision to disallow Mehdi Taremi's late goal.
Rodrygo snatched victory for Real as he flicked in with a minute to play in Milan, stunning Inter who had dominated much of the game and forced Thibaut Courtois into several saves.
Newcomers Sheriff defeated Shakhtar Donetsk 2-0 thanks to goals from Mali international Adama Traore and Guinean forward Momo Yansane.
Sheriff, from a tiny pro-Russian separatist region in Moldova, are the first club from the ex-Soviet country to reach the group stage of the Champions League.
England midfielder Jude Bellingham scored one goal and set up Erling Braut Haaland for Borussia Dortmund's second in a 2-1 win away to Turkish champions Besiktas.
Javi Montero headed in a consolation for Besiktas in stoppage time.
Ajax Amsterdam powered to the top of Group D as Ivorian striker Sebastien Haller bagged four goals in a 5-1 demolition of Portuguese champions Sporting Lisbon.
Steven Berghuis added the other goal for the four-time European champions with Paulinho grabbing a consolation for the hosts.
Brugge 1 (Vanaken 27) PSG 1 (Herrera 15)
Man City 6 (Ake 16, Mukiele 28-og, Mahrez 45+2-P, Grealish 56, Cancelo 75, Jesus 85) Leipzig 3 (Nkunku 42, 51, 73)
Atletico 0 Porto 0
Liverpool 3 (Tomori 9-og, Salah 49, Henderson 69) Milan 2 (Rebic 42, Brahim 44)
Besiktas 1 (Montero 90+4) Dortmund 2 (Bellingham 20, Haaland 45+3)
Sporting 1 (Paulinho 33) Ajax 5 (Haller 2, 9, 51, 63, Berghuis 39)
Sheriff 2 (Traore 16, Yansane 62) Shakhtar 0
Inter 0 Real Madrid 1 (Rodrygo 90)