Wayne Rooney proved he is back to his best following a summer of upheaval as he scored twice to help Manchester United to a 4-2 victory over Bayer Leverkusen and Shakhtar Donetsk won 2-0 at Real Sociedad.
In Group C, Paris Saint-Germain crushed Olympiakos 4-1 and Benfica beat Anderlecht 2-0.
Bayern, which appointed Pep Guardiola as coach after claiming an unprecedented treble last season, was rarely troubled by the Russian league leader.
Alaba gave the German champions an early lead with a free kick into the left side of the goal after being fouled on the edge of the penalty area. Mandzukic, who had earlier hit the post, headed home Robben's in-swinging free kick to double Bayern's advantage in the 41st minute.
Robben sealed the result in the 68th, blasting home Alaba's ball over the defense.
Bayern is attempting to become the first club since 1990 to defend the European Cup.
Manchester City matched Bayern thanks to second-half goals from Edin Dzeko, Yaya Toure and Sergio Aguero.
Real Madrid is bidding for a first Champions League trophy since 2002, and it sent out a warning sign to its rivals with a resounding victory in Turkey.
"The match was good and the result fantastic," coach Carlo Ancelotti said. "After Isco's goal the match became simpler. We defended well and when Benzema and Cristian Ronaldo have a good night on the counterattack, it's easy, even though winning in this stadium is not easy."
There was plenty of attention for Gareth Bale as he made his Champions League debut for Madrid in the second half. But it was Ronaldo the player Bale usurped as the world's most expensive footballer who was the star of the night.
Francisco "Isco" Alarcon opened the scoring after 33 minutes and Karim Benzema went on to strike twice during a second half in which Ronaldo was unstoppable.
The former Manchester United player scored Madrid's third in the 63rd and doubled his tally three minutes later. Ronaldo completed his hat trick in stoppage time, capping a brilliant night for him and his team by weaving past three Galatasaray defenders before smashing the ball into the back of the net.
It was Ronaldo's second Champions League hat trick and a fitting way for him to celebrate signing a new contract with Madrid.
Juventus will have a hard time beating Madrid to top spot in the group and it got off to a difficult start in Denmark.
Antonio Conte's side was stunned when Nicolai Jorgensen fired Copenhagen in front in the 14th minute, beating Italy goalkeeper Gianluigi Buffon from close range after Pierre Bengtsson's free kick.
Fabio Quagliarella leveled nine minutes after halftime.
Manchester United manager Moyes has limited experience in the competition, having lost both qualifying matches back in 2005 when in charge of Everton.
Rooney, still wearing a thick, black headband to cover a deep gash to his forehead sustained in a training ground collision, opened the scoring in the 22nd minute. Simon Rolfes equalized against the run of play for Leverkusen in the 54th but Robin van Persie restored United's lead five minutes later.
Rooney then scored his 200th goal for United in all competitions before setting up Valencia following a quick counterattack for the winger to fizz a low shot in at the near post.
"I've concentrated on working hard to get my head down, to get myself fit and ready for the season," Rooney said. "I'm delighted with the way I've come back and delighted I'm back playing and scoring goals.
"It's great the reception I get here and hopefully I can repay (the fans) with performances and goals like tonight."
Omer Toprak poked in a close-range consolation goal for Leverkusen two minutes from time.
Alex Teixeira struck twice in the second half to give Shakhtar Donetsk a 2-0 victory over Real Sociedad in Group A.
In Greece, Edinson Cavani opened the scoring for PSG in the 19th minute but Vladimir Weiss leveled six minutes later, weaving past three defenders for an impressive solo effort.
However, Thiago Motta headed in two second-half goals and Marquinios completed the scoring in the 86th. Zlatan Ibrahimovic missed a late penalty for the French champions.
Filip Djuricic and Luisao scored goals in the first half-hour to help Benfica to victory.
