Eintracht's Ritsu Doan then fired over the bar and teammate Fares Chaibi's spot-kick was saved by Gregor Kobel as Dortmund kept their cool and scored all four of their penalties.
The visitors had an early chance with Karim Adeyemi's shot but in the next move Frankfurt's Ansgar Knauff slotted in from a superb through ball by Mario Goetze, as the two former Dortmund players combined to put the hosts ahead in the seventh minute.
Dortmund, who visit Manchester City in the Champions League next week, shot out of the blocks after the break with Julian Brandt tapping in at the far post for a 48th-minute equaliser.
Dortmund coach Niko Kovac, who led Frankfurt to German Cup success in 2018, brought their top striker Serhou Guirassy off the bench with 13 minutes left. It was the home side, however, who came close again with Doan rattling the bar in the 81st.
Eintracht thought they had won it with Jonathan Burkardt's close-range effort in the 115th but he was offside before they were beaten in the penalty shootout.
