Erling Haaland (22), who scored City's third goal of the night in the 76th minute, has 45 goals in all competitions, a single-season record for a Premier League player, moving ahead of Mo Salah (30) and Ruud van Nistelrooy (46).
Rodri (26) put Pep Guardiola's team ahead in the 27th minute, cutting the ball back to beat midfielder Jamal Musiala (20) and curling a blistering long-range shot into the top left corner beyond the outstretched hand of Yann Sommer (34).
"I know that this Bayern team is special," said Guardiola, who managed Bayern from 2013-16. "We got a good result, but we need two strong games. Anything can happen at Allianz Arena. I know that because I was at Bayern for three years."
Rodri celebrated with a long knee slide on the rain-soaked turf at the Etihad, while Bayern's new manager Thomas Tuchel threw up his hands in disgust en route to his team's biggest defeat in the Champions League since a 3-0 loss to Paris St Germain in 2017.
"It was very special for me to score my first goal in the Champions League. At that moment we were struggling because they are a great, great team," Rodri told BT Sport.
"Sometimes we understand we cannot have 80% possession. They play too. We scored three and could score four but it is a great result."
"This never happens to Bayern Munich but credit to Manchester City, they were brilliant in both boxes," former Bayern player Owen Hargreaves said on BT Sport. "Pep had a game plan and it worked beautifully."
Guardiola faced a familiar foe in Tuchel, who was Chelsea manager when they triumphed over City in the final in 2021.
The Spaniard has won four Premier League titles, one FA Cup and four League Cups at City but European success has eluded the manager who won the trophy twice when in charge of Barcelona.
His confidence will be high that he can end that drought ahead of the second leg, which takes place on April 19 at Munich's Allianz Arena.
"We don't think it is done. They are a good team and we are happy with the performance but we will have to try to win the game," Rodri added.
