American teenager Christian Pulisic became the fourth youngest player to score in the Bundesliga when the 17-year-old drilled in from a tight angle in the 38th minute.
Adrian Ramos added another goal six minutes later, shaking off two defenders to fire in as Dortmund looked to recover from their last-gasp Europa League quarter-final elimination by Liverpool last Thursday.
"I can't describe it, can't put it into words," a beaming Pulisic told reporters, after also earning praise from U.S. national team coach Juergen Klinsmann. "I don't think I'll be sleeping much tonight."
"Obviously, we were disappointed (after the Liverpool defeat)," said the U.S. international. "But our coach told us we have to keep going and not hang our heads. It did us a lot of good."
Hamburg had keeper Rene Adler sent off for bringing down a charging Shinji Kagawa after the break.
Hamburg played the final 13 minutes with nine men after Alvin Ekdal also left the field injured with no substitutions available.
Dortmund's numerical advantage meant the hosts only needed to play possession football as Hamburg sought to limit the damage. But Colombian Ramos struck again with his eighth goal of the campaign in the 86th minute.
The win lifted Dortmund, who have already secured a Champions League spot, to 71 points, seven adrift of Bayern, 3-0 conquerors of Schalke 04.
European hopefuls Mainz 05 squandered a two-goal lead and a chance to close on the Champions League spots, slumping to a 3-2 loss against Cologne, who scored three times in the second half to move up to 10th on 37. Mainz are sixth on 45, level with seventh-placed Schalke 04.
(Reporting by Karolos Grohmann, editing by Ed Osmond and Neil Robinson)
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