The league's top scorer, who also scored one goal and set up another last week, struck twice in seven minutes early in the second half to take his tally to 14 as the Bavarians won for the 16th time in 20 games.
Thomas Muller had given the visitors a 40th-minute lead against the run of play with his 11th league goal of the season.
Bayern, who have won nine of their 10 away games this season without conceding a goal, have 51 points.
"We knew it would be a hard game and we prepared very well for it," Bayern coach Jupp Heynckes told reporters.
"We showed that we have fun playing football. This was a big win towards the title," said the German who will make way for former Barcelona coach Pep Guardiola next season.
Eintracht Frankfurt moved level with third-placed Dortmund on 36 points after Srdjan Lakic, signed on loan from VfL Wolfsburg this week, made a dream start by claiming both goals in a 2-0 victory at Hamburg SV.
Sixth-placed Schalke 04's new signing Michel Bastos scored on his debut but it was not enough as the Champions League last 16 qualifiers slumped to a 2-1 home defeat by bottom club Greuther Fuerth thanks to Nikola Djurdjic's last-gasp goal.
It was Fuerth's second league win of the season, and their first since August.
Mainz, fifth on 30 points, were the livelier side at the start as they took the sting out of Bayern's game with incessant pressure and quick breaks.
The home team twice went close in the first half through Shawn Parker and Yunus Malli.
Bayern took the lead however when Muller beat the offside trap as he chased a Toni Kroos pass and rounded keeper Christian Wetklo to slot the ball in from a tight angle.
Mandzukic, again picked to start ahead of Mario Gomez, added the second goal five minutes after the restart when he tapped in a low David Alaba cross following the Austrian's run down the left.
With Mainz now trying to play catch-up, Bayern had far more space and it did not take long before they made it count.
Muller sent captain Philipp Lahm through on the right and his cross was met by Mandzukic as he headed his second goal at the far post in the 57th minute.
Fortuna Dusseldorf stunned VfB Stuttgart 3-1 with two goals by Australian Robbie Kruse as they rose to 13th place and away from the drop zone.
Hoffenheim climbed one spot to third from bottom courtesy of a 2-1 win over Freiburg, former Tottenham Hotspur keeper Heurelho Gomes making a triumphant debut.
Second from bottom Augsburg continued their improved start to the year, chalking up a third undefeated game in a row with a 1-1 draw at VfL Wolfsburg.
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Try your first month for just £1 / $1 / €1
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.