With new signing Joao Cancelo in the starting lineup 24 hours after penning his loan deal from Manchester City, Bayern played a stellar first half, scoring three times to kill off the tie quickly.
The German champions, who face Paris St Germain in the Champions League Round of 16 in two weeks, had started the year with three consecutive draws in the Bundesliga.
Eric Maxim Choupo-Moting volleyed in at the far post to give Bayern the lead in the 17th minute before Jamal Musiala doubled their advantage on the half-hour mark with a low left-footed drive that went in off the post.
Bayern kept up their intensity level and hit the woodwork with a Joshua Kimmich free kick before scoring again two minutes before the break when Choupo-Moting flicked on a high ball to Leroy Sane who had no trouble slotting it in from eight metres.
The visitors eased off after the break and Mainz found more space but they were beaten once more when Alphonso Davies rose high to power in a header for Bayern's fourth goal in the 83rd minute.
The hosts deserved at least a consolation goal but instead saw coach Bo Svensson sent off for dissent before defender Alexander Hack also got his marching orders in the 87th for a second booking.
Holders RB Leipzig also moved into the next round with a 3-1 victory over Hoffenheim.
