With Dortmund not in action against Bayer Leverkusen until Sunday, the champions seized the opportunity to pile pressure onto their rivals despite a largely poor performance against Hertha.
The visitors carried the greater threat before Javi Martinez nodded Bayern in front just after the hour, settling nerves among the home supporters.
The final 30 minutes were still not wholly convincing, however, and long-suffering Bayern winger Kingsley Coman was dealt another injury blow, disappearing down the tunnel nine minutes after arriving from the bench.
A frustrating afternoon for a Hertha side who would have been good value for at least a point ended with Karim Rekik sent off as he lashed out at Robert Lewandowski off the ball.
Bayern lacked fluency in the first half and saw an early offside flag deny the visitors' Davie Selke the opener.
Jerome Boateng, in for the ill Mats Hummels, was required to hack clear in front of the goal line when Salomon Kalou beat Manuel Neuer.
There was initially little improvement from the hosts following the restart and only a superb Joshua Kimmich block diverted a goalbound Selke strike past the left-hand upright.
But it took just one opportunity for Bayern to break the deadlock, with the unlikely figure of Martinez making the difference as he rose above Rune Jarstein to head home.
There were few further chances, with only Coman's latest setback of real note as his team-mates had their task of holding on made easier by Rekik.
What does it mean?
Was this the week Kovac found his feet at Bayern? After a tough start to life at the Allianz Arena, Kovac has overseen significant progress at home and abroad in recent days. A goalless draw at Liverpool gives the Bundesliga giants every chance of Champions League progression, while the pressure is now on Dortmund in the league as they face Leverkusen.
Martinez makes the difference
Spanish midfielder Martinez was lauded for his display against Liverpool and he was again an assured presence in the middle of the field on Saturday. But performances matter little at this stage of a title chase and it was Martinez's goal that made all the difference for Bayern.
Wingers fail to deliver
That Bayern were so flat for so long had little to do with Robert Lewandowski, who was completely isolated. Franck Ribery offered little before he was substituted, while Serge Gnabry also failed to service the front man. Coman's afternoon was perhaps even more difficult, starting from the bench and then bowing out with injury.
Key Opta facts
- Kingsley Coman appeared only for nine minutes and five seconds, but had more chances created (one) than Franck Ribery, whom he replaced, in 58 minutes.
- Javi Martinez scored his first home goal since April 2017 (versus Borussia Dortmund). He has scored five of his last six competitive goals following a corner.
- For the first time since May 2009, Thomas Muller was benched in four successive competitive games for which he was eligible.