Achraf Hakimi's 113th minute strike appeared to have clinched Dortmund's place in the quarter-finals but Bremen's Martin Harnik netted six minutes later to force the shootout.
Bremen, who are 10th in the league standings, took a fifth minute lead through Milot Rashica but Dortmund captain Marco Reus levelled in first-half stoppage time from a free kick. Reus had to be substituted due to injury after the interval.
U.S international Christian Pulisic put the hosts 2-1 ahead at the end of the first half of extra time but Claudio Pizarro levelled for Bremen after the second interval, leaving Hakimi and Harnik to trade blows for each side before the shootout.
(Reporting by Richard Martin; editing by Christian Radnedge and Ken Ferris)
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