Czech champion Viktoria Plzen lost 3-0 at home to Manchester City in the other Group D match.
Bayern is attempting to become the first club since 1990 to defend the European title. The Bavarian side defeated domestic rival Borussia Dortmund in the Champions League final in London last May.
"I could hear from fans today, 'Finale, Finale,' but let's keep out feet on the ground. We need to go step by step," Robben said. "It's early in the season. The most important thing is to concentrate on ourselves, improve as we go on. It was a good start."
Bayern sports director Mattias Sammer had criticized his side's players after a 2-0 Bundesliga win over Hannover on Saturday for not delivering their full potential.
"He's entitled to criticize us," Alaba said after his impressive performance.
"That had nothing to do with Mattias Sammer," said Bayern chairman Karl-Heinz Rummenigge after the win. "The team has huge quality, the team has our trust. At the end of the day we know that they can deliver such games. They started where they stopped in London. Namely, with a class game."
Alaba's free kick rewarded the home side for its bright start. The 21-year-old Alaba, who scored the winner for Austria over Ireland in their World Cup qualifier last week, dusted himself off after being fouled on the edge of the penalty area and thumped a powerful shot over the wall and inside the left post.
Mandzukic should have made it 2-0 soon after when he was one-on-one with the goalkeeper but Igor Akinfeev managed to block the shot.
Nigerian striker Ahmed Musa shot wide for Moscow's first chance on a counterattack in the 12th, before Bayern regained control.
Mandzukic missed opportunities either side of a Rafinha shot that struck the side netting, his second chance narrowly missing when his header struck the post and bounced in front of the line.
The Croatian player finally scored when he was one of three Bayern players queuing up to meet Robben's in-swinging free kick. At least two of the players appeared to be offside according to TV replays.
There was improvement from the visitors after the break, but any hope the Russian club had of getting back into the game was killed when Alaba lobbed the ball over the defence, allowing Robben time to control the ball on his chest before blasting past Akinfeev from close range.
"We played well, especially in the first half," Guardiola said. "I'm very proud. The first game back after winning the Champions League is never easy."
