Substitute Pierre-Emerick Aubameyang converted a penalty after 1860's Dominik Stahl was sent off for a foul on Marco Reus in the 105th minute and Henrikh Mkhitharyan added another after a textbook break two minutes later.
Dortmund were kept at bay in the first half by a well-oiled 1860 defence before the visitors rattled the crossbar with a Jakub Blaszczykowski effort right after the restart.
Dortmund, eager to rebound from their opening Champions League defeat to Napoli and a 1-1 draw at Nuremberg last week, upped the pressure and had more than 70 percent possession in the second half but could not work their way into the box.
Dortmund, Cup winners in 2012, were also repeatedly denied by 1860's veteran keeper Gabor Kiraly, who at 37 showed he had lost none of his reflexes when he flicked a Marco Reus shot onto the post with his foot early in extra time.
Germany international Reus then rattled the bar in the 97th, curling a free kick over the wall before earning a penalty because of a push by Stahl, who was given his marching orders.
Gabon international Aubameyang sent Kiraly the wrong way to break the deadlock after his team had 32 shots on target compared to two for their opponents.
With 1860 down to ten men and with tired legs having made no substitutions, Dortmund struck again two minutes later after a dazzling move.
Keeper Mitchell Langerak fed Jonas Hofmann, who sprinted down the right, cut into the box at just the right moment and fed the Armenian forward to round Kiraly, beat a defender and slot in. Aubameyang still had time to whip a free kick against the bar with three minutes left to play.
Augsburg enjoyed a much easier evening, cruising past third tier Preussen Muenster 3-0 while fellow Bundesliga club VfL Wolfsburg also booked their last-16 spot with a 2-0 win over second-division Aalen.
Holders Bayern Munich take on Hanover 96 in an all-Bundesliga second round match on Wednesday.