Fellow Bundesliga club Schalke, however, crashed out in a 2-1 defeat to third-tier Dynamo Dresden.
Former Germany international Adler came to the rescue after Hamburg had failed to break the resistance of their lower-ranked opponents in a 2-2 draw after 120 minutes.
The former European champions, who needed a relegation playoff to remain in the Bundesliga last season, trailed 1-0 from the 10th minute until Heiko Westermann levelled an hour later.
The Bundesliga club thought they had finally done enough to win it when Dutchman Rafael van der Vaart, who had set up the equaliser, curled in a free kick in the first half of extra time.
Cottbus' Sven Michel levelled in the 105th minute to take the game to penalties, where Adler saved Michel's spot kick and that of Fabian Pawela to save Hamburg's blushes.
Schalke, who will play in the Champions League this season, tumbled out of the competition at the first hurdle against Dresden with a poor performance.
A goal in either half gave the hosts, banned from the competition last season due to repeated crowd trouble, a two-goal lead before a 78th minute Joel Matip goal gave Schalke hope.
Julian Draxler tried to force the equaliser by dribbling past three players in stoppage time, but he saw his effort saved and Dresden celebrated a winning return to the competition.
Holders Bayern Munich advanced comfortably 4-1 against Preussen Muenster on Sunday to join last season's runners-up, Borussia Dortmund, in the second round.
(Reporting by Karolos Grohmann,; editing by Justin Palmer and Stephen Wood)
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