Hamburg remain second last, three points below Mainz, who lost 4-2 at Hoffenheim.
Pulisic missed the target from 15 yards out in the 64th minute as Dortmund went in search of more goals, and the United States of America worldwide was a whisker away from getting on the end of Mahmoud Dahoud's cross 13 minutes later.
Dortmund lived dangerously with Hamburg threatening an equalizer, but substitute Mario Gotze sealed the win in injury time after Andre Schurrle played him in to lift the ball over the onrushing goalkeeper. Hamburg defended well and looked more like scoring.
Dortmund found it hard to create opportunities and it was the hardy visitors who registered the first - and only - shot on target in the opening period, Filip Kostic's cross-goal effort comfortably held by Burki five minutes before the break. We've waited a long time and we hope that he stays healthy.
Congestion remains the name of the game as the Bundesliga's top teams duke it out for Champions League places, though familiar faces sit 1-2-3 after wins from RB Leipzig and Borussia Dortmund on Saturday.
However, the on-loan Chelsea striker took his next chance early in the second half as he capitalised on a miscued shot from Christian Pulisic to slot home at the far post for his third goal in the black and yellow shirt.
However, the Germany forward was unable to inspire Dortmund in a lackluster first-half.
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