Dortmund, who were reduced to 10 men late in extra time after Sokratis Papastathopoulos was sent off for dissent, were better from the penalty spot as three Hertha players failed to score.
Dortmund's Pierre-Emerick Aubameyang and Hertha's Vedad Ibisevic both had golden chances in the first half, before Salomon Kalou volleyed Hertha ahead in the 27th minute.
Marco Reus drew the hosts level three minutes into the second half after Ousmane Dembele's shot bounced off the post.
In a one-sided second half Dortmund should have won it after creating several clear scoring chances.
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They were reduced to 10 men when Papastathopoulos was booked twice in quick succession, leaving the pitch in the 119th minute.
The hosts, however, kept their cool in the shootout before Kalou fired his penalty high over the bar to send Dortmund into the next round.
Third division Sportfreunde Lotte continued their sensational run by beating second tier 1860 Munich 2-0 to advance to the last eight. Lotte have already eliminated Werder Bremen and Bayer Leverkusen.
Schalke 04 eased past second division Sandhausen 4-1 while holders Bayern Munich edged past VfL Wolfsburg 1-0 to book their spot on Tuesday.
(Reporting by Karolos Grohmann; Editing by Toby Davis)
