The Bavarians, chasing a fifth straight Bundesliga title, lacked any spark for much of the game, with Tuesday's Champions League quarter-final extra-time loss to Real Madrid having taken its toll on the players.
They are on 70 points, with second-placed RB Leipzig on 61, ahead of their match at Schalke 04 on Sunday.
"It was difficult for us but we expected more," said Bayern coach Carlo Ancelotti. "We were not as compact in the first half. We should have done it better.
"We were better after the break and luckily managed to get the equaliser."
Bayern, who next face Borussia Dortmund in the German Cup semi-final next week, made an error-ridden start and Bojan Krkic sneaked in to fire the visitors ahead after three minutes.
Levin Oztunali could have doubled the lead from close range a little later as Bayern looked disorganised.
Arjen Robben levelled after a fine break from the Bavarians but they were again on the back foot when Mainz were awarded a 40th-minute penalty and Daniel Brosinski beat goalkeeper Sven Ulreich.
Thiago Alcantara's low drive from the edge of the box drew Bayern level for the second time but despite late efforts and more than 70 percent of possession they could not find a winner.
They have now gone three matches in all competitions without a victory.
Raphael Guerreiro's 87th-minute glancing header gave Borussia Dortmund a 3-2 victory at Borussia Moenchengladbach to take over third place and the last spot leading to automatic Champions League group-stage qualification.
Dortmund, eliminated by Monaco from the Champions League this week, are on 56, a point ahead of Hoffenheim, who drew 1-1 at Cologne on Friday.
Vedad Ibisevic's 12th goal of the season gave Hertha Berlin a 1-0 win over VfL Wolfsburg to stay on course for a Europa League spot in fifth place on 46.
Max Kruse scored four times in 46 minutes as Werder Bremen crushed Ingolstadt 4-2 to extend their unbeaten run to 10 matches and climb to sixth on 42 points as they continue their remarkable revival since the winter break.
Last-placed Darmstadt 98 earned themselves a mathematical lifeline with a 2-1 win at Hamburg SV but they are still 11 points off the relegation playoff spot with four games left.
(Reporting by Karolos Grohmann; Editing by Toby Davis and Clare Fallon)
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