The Bavarians, seven points behind Dortmund a few weeks ago, have now won 10 of their last 11 league games to draw level on 51 points but Bayern stay second on goal difference and Dortmund could reclaim the outright lead on Sunday if they avoid defeat when they host an in-form Bayer Leverkusen.
Bayern, who drew 0-0 at Liverpool earlier this week in their Champions League round-of-16 first leg, were in complete control but lacked clinical finishing as Hertha waited to counter-attack.
"Today we won a difficult game and it was all about the victory," said Bayern coach Niko Kovac.
"We had problems in the first half and at the end of the day it was a set piece that decided this game. We had also worked very hard in Liverpool in midweek and the lads now deserve two days' rest. Overall I am satisfied with the performance."
Bayern's Joshua Kimmich almost scored an own goal when he attempted to clear a Davie Selke effort on the line only to mis-hit the ball, which luckily rolled wide, in the 57th.
Five minutes after Hertha's biggest chance, however, Spaniard Martinez headed the hosts in front from a James Rodriguez corner to settle their nerves.
Hertha were left with 10 men when Karim Rekik lost his temper and hit striker Robert Lewandowski in the 84th minute following a challenge outside the Hertha box.
There was also bad news for Bayern, with winger Kingsley Coman, recently recovered from an ankle injury, being taken off with a suspected thigh injury nine minutes after coming on in the 58th.
"We will have to wait for further checks to see about Kingsley Coman," said Kovac. "I hope it is not too bad."
Third-placed Borussia Moenchengladbach lost further ground in the title race with a 3-0 home loss to VfL Wolfsburg, their third straight game without a win that left them stuck on 43 points.
Schalke 04 slumped to a 3-0 defeat against Mainz 05, days after their spirited performance in a 3-2 defeat to Manchester City in the Champions League, to remain in 14th and trigger the early departure of sports director Christian Heidel.
He informed the club after the loss that he would leave by the summer at the latest, a year ahead of his contract's end.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar and Clare Fallon)
