After the €63.75million departure of the prolific Pierre-Emerick Aubameyang to Arsenal, the on-loan Chelsea striker's form and Reus' first appearance since last May having been sidelined through anterior cruciate ligament damage will help raise sprits at Signal Iduna Park.
Hamburg remains second last, three points below Mainz, which lost 4-2 at Hoffenheim.
Reus showed no signs of rust during the 71 minutes of his comeback, but - despite an attacking line-up also including Andre Schurrle, Christian Pulisic and Shinji Kagawa - Dortmund found the going tough against Bernd Hollerbach's well-drilled side.
As promised, the Belgian celebrated with a somersault after scoring on his home debut.
Reus started Dortmund's game against Hamburg at the Westfalenstadion and was handed the captain's armband.
'Catastrophe' star Rob Delaney's 2-year-old son dies of cancer
I don't want to miss out on their handsome lives. "I'm greedy for more experiences with them", he wrote. Much obliged to you, wonderful Henry , for investing as much energy with us as you did.
More comical defending led to Berggreen poking in a second nine minutes from time only for Kramaric to blast in a superb free kick and send the hosts up to eighth in a bunched top half.
Goals on the day came from new boy Michy Batshuayi and Mario Gotze, as the win lifts Dortmund into third place following Bayer Leverkusen's surprise defeat to Hertha Berlin. There were whistles from disappointed Dortmund fans at the break. But Bayern regained the lead seven minutes later with a crafty effort from Mueller who got to the byline, saw that Faehrmann was expecting him to pull the ball back and threaded his shot between the goalkeeper and the near post.
A goal still looked likely when Dahoud's wayward shot found Pulisic in space in the six-yard box, but the United States global somehow aimed high of the gaping goal.
An ankle injury suffered in the final warm-up game before the 2014 World Cup in Brazil ruled the playmaker out of Germany's title-winning squad.
HSV were superb in the first half to belie their disappointing league standing as Dortmund failed to register a shot on target before half-time.