Bayern recovered well from the early shock but went further behind when Mbappe held the ball up for Edinson Cavani to hit a side-footed shot from the edge of the area into the roof of the net.
The third followed in the 63rd minute with a strong run through midfield by Alves and some neat play in the box by Mbappe whose cross was blocked by Javi Martinez, only for Neymar to pounce at close range.
Celtic, beaten 5-0 at home by PSG in their campaign opener, picked up full points with a 3-0 win in Anderlecht courtesy goals either side of half-time by Leigh Griffiths and Patrick Roberts before a late strike from Scott Sinclair.
In Moscow, Lukaku put United ahead against CSKA with a fourth-minute header and made it three in the 27th following an Anthony Martial penalty.
Henrikh Mkhitaryan netted a fourth in the 57th minute, with the home side scoring a late consolation through Konstantin Kuchaev.
United top Group A on six points, while Basel draw level on three points with CSKA by thrashing Benfica 5-0.
Early goals from Michael Lang and Dimitri Oberlin were followed up by a 59th- minute penalty from Ricky van Wolfswinkel.
Benfica were down to 10 men four minutes later when Andre Almeida was shown a red card for a lunging tackle, and Basel completed the rout with a second from Oberlin and one from Blas Riveros.
Chelsea enjoyed a strong first half in Madrid but it was Atletico who went in at half-time ahead after David Luiz pulled the shirt of Lucas Hernandez at a corner, and Griezmann converted from the spot five minutes before the break.
The London side levelled on the hour when Alvaro Morata met Eden Hazard's cross with a glancing header, and Michy Batshuayi struck a stoppage-time winner in front of goal with the last kick of the game.
Roma defeated Azerbaijan champions Qarabag in Baku to go to four points, two behind Group C leaders Chelsea.
Goals from Kostas Manolas in the seventh minute and Edin Dzeko in the 15th put Roma in control but the home side fought back with Pedro Henrique scoring in the 28th for Qarabag's first-ever Champions League goal.
However, the Serie A side held on for their first Champions League win in seven attempts and their first away from home since 2010.
An own goal by Sporting Lisbon's Sebastian Coates, who deflected in a Luis Suarez header, was enough for visiting Barcelona to earn a second straight win in Group D.
Juventus picked up their first points with a 2-0 victory over visiting Olympiakos. Gonzalo Higuain broke the deadlock in the 69th minute and Mario Mandzukic followed up with a header 11 minutes later.
- AAP
