To add insult to injury for City manager Roberto Mancini, Argentine striker Carlos Tevez refused to come on as a second-half substitute and looks set to never play for the club again.
Bundesliga top scorer Gomez, who only returned at the weekend from a groin injury, snatched a 38th-minute lead when he slid in after City keeper Joe Hart had saved efforts from Franck Ribery and Thomas Muller.
"In the first 25 minutes we saw that City are a very good team and they are top of the Premier League alongside Manchester United for a reason. We needed our time to get into the game," Bayern coach Jupp Heynckes told reporters.
"We gradually improved, used weaknesses in our opponents' game and we showed how homogenous a team we are. To again keep a clean sheet especially in this competition means we are in very good form and my players deserve compliments."
With only one win in 17 appearances by English teams in Munich, the odds were stacked against City but the visitors looked livelier at the start and threatened after only a few minutes when Edin Dzeko's low drive was blocked by goalkeeper Manuel Neuer.
Mancini opted to start with the blossoming attacking partnership of Dzeko and Sergio Aguero, who between them have scored 14 league goals in City's unbeaten start to the Premier League, leaving Tevez on the bench.
Gradually, however, Ribery carved out space down the left wing and it was the Frenchman who set up the first goal, firing in from outside the box, keeper Joe Hart doing well to save it as well as Thomas Muller's rebound before the unmarked Gomez slid in for the lead.
On the stroke of half-time Toni Kroos floated in a free-kick, Hart again was on hand to save Daniel van Buyten's glancing header but was helpless when Gomez scored from close-range, leaving the keeper furious with his defenders.
Mancini was furious with his strikers, taking off Dzeko and saying he was unhappy with the Bosnian's performance.
"I think we need to improve," Mancini told reporters. "We played well for 25 minutes, 30 minutes but we conceded a stupid goal.
"But if we want to improve as a team Carlos can't play with us. With me, he is finished."
