Heiko Westermann and Rafael van der Vaart both found the net for Hamburg as the game ended 2-2 after extra time, before they prevailed 4-1 on penalties against their third-tier hosts.
Schalke, though, were unable to join their Bundesliga rivals as they lost 2-1 to third-tier Dynamo Dresden even with stars such as Kevin-Prince Boateng and Klaas-Jan Huntelaar in their side.
Their travails were in contrast to holders Bayern Munich, who eased through, while Hoffenheim recorded a 9-0 victory against fifth-tier amateurs Paloma.
Schalke’s fellow top-flight sides Mainz, VfB Stuttgart and Augsburg — beaten by Dresden’s fellow former East German rivals Magdeburg — were also eliminated at the first hurdle.
The new Bundesliga season begins on Friday, when Bayern Munich start their defence of the title against Wolfsburg.