Bruno Labbadia's appointment in midweek failed to provide a spark for Hamburg, who slipped to a club-record sixth successive game without scoring and are without a win since February 7.
There was little to get excited about from Hamburg's perspective at the Weserstadion, with opportunities something of a rarity throughout.
Franco Di Santo scored the game's only goal, converting a penalty kick six minutes from time after Valon Behrami was dismissed for hauling down Zlatko Junuzovic inside the area.
Hamburg failed to mount a response afterwards and ultimately tasted another defeat to leave them four points adrift of guaranteed safety.
In the day's late kick-off, Wolfsburg needed a late equaliser to salvage a 1-1 draw at home to Schalke, who were denied a first win in five.
After a goalless first half, 19-year-old Leroy Sane put Schalke ahead with his third goal of the season in the 53rd minute, making the most of a kind ricochet before slotting past Diego Benaglio.
Wolfsburg's pressure paid off eventually, however, and it came as little surprise to see Kevin De Bruyne involved, as the Belgian found the bottom corner from the edge of the area with 12 minutes to go.
The hosts could not find an elusive winner, though, and the result means Bayern Munich need only four points from their remaining five games to clinch a 25th Bundesliga title.
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