Muller scored just before half-time to give die Roten the lead, before Croatian striker Mario Mandzukic wrapped up the game with two second-half goals.
The Germany attacker told reporters after the game: "We are delighted to have won such a difficult away match 3-0. We knew we had to work hard [in the match]."
"In the first half, Mainz were very aggressive as they were on home soil and made life difficult for us as we had expected.
"Going 1-0 up was quite important. From there, we had the opportunity to go on and beat Mainz. In the second half, we flew out of the traps and soon the match was over. It was a pretty good day."Bayern coach Jupp Heynckes was also pleased with the outcome, and stressed how crucial the triumph could be for the Bavarian club.
"We knew that it would not be an easy game, and we were very diligent [in our victory]," he said. "The team performed very well. Today was a very important win for us towards the Bundesliga title."