Ousmane Dembélé also scored for BVB, with Nicolai Müller's brace proving the only source of hope for the home side.
Markus Gisdol knew change was necessary as HSV's winless run continued last weekend, and he made five alterations as a result. Michael Gregoritsch, Müller, Emir Spahic, Cleber and Dennis Diekmeier came in for Aston Götz, Filip Kostic, Lewis Holtby, Matthias Ostrzolek and Bobby Wood.
Aubameyang, Sebastian Rode and Emre Mor were brought in from the 1-0 win over Sporting CP. That meant Thomas Tuchel afforded a rare rest for Julian Weigl, as Mario Götze and Dembélé dropped to the bench.
Aubameyang hits first-half hat-trick
After his well-documented absence in mid-week, it didn't take long for Aubameyang to get back in Tuchel's good books. A driving run and cross from Christian Pulisic found Mor arriving late in the box. The youngster's shot was parried poorly by René Adler, and straight to Aubameyang; he tapped home from four yards to open the scoring.
Unfortunately, Pulisic's run would prove to be the only moment of quality in the opening 20 minutes of the match. It was scrappy and scruffy, with misplaced passes and niggly tackles dominating the early exchanges. Pierre-Michel Lasogga was a lonely figure up front, as HSV struggled to break out of their own half at the best of times.
To make matters worse, an already disgruntled home crowd were livid when Johan Djourou turned and gave up possession to Mor with an horrific back pass. He was only too happy to run through on goal and was joined by Aubameyang, and Mor slid the ball across to the Gabonese forward to tap into an empty goal.
Livid turned to toxic four minutes later as Aubameyang rounded off his hat-trick. This one required more composure than his first two, as Marc Bartra's clearing header went over the HSV back-line and into Aubameyang's path. He went one-on-one with Adler and fired low, with the 'keeper disappointed not to do better as it squirmed in.
To their credit, Hamburg managed to stem the flow before the break and kept BVB reasonably quiet. They even began to work forward themselves as headed chances seemed to be the best route back into the game. However, Roman Bürki was very rarely troubled in the visiting goal as Dortmund went into the break with a 0-3 lead.
Hamburg mount a fightback but fall short
HSV came out with hope that things couldn't get any worse, and a promising start looked to back that up. However, Aubameyang had other ideas. Pulisic pounced after Spahic ad Djourou doddled on the ball, shifting it right to the man with the hat-trick and he made no mistake in rifling home from 18 yards into the bottom corner.
BVB could have scored twice after that but instead gave up a goal to their extremely high line. Gian-Luca Waldschmidt had only just been brought on by Gisdol, but had the presence of mind to flick the ball over the stranded visiting defence and play in Müller. He ran through on goal, finishing past Bürki with aplomb.
When it rains it pours, and being bottom of the league Hamburg could do with a decision going their way. Unfortunately for them, Sascha Stegemann decided that Gregoritsch had impeded Matthias Ginter at the back post as he tapped in Diekmeier's cross. It was most certainly a debatable call from the official.
That seemed to take the stuffing out of HSV who were limited, largely, to pot-shots from long range. Instead it was Tuchel's charges that grabbed another goal and it was Dembélé who grabbed it just moments after coming off the bench. The Frenchman stayed composed to curl past Adler from close range.
Yet Hamburg and Müller would come again. He sent a screaming shot into the bottom corner from 25 yards with great pace and accuracy, as he proved to be their only bright spark on a dull day that seen them stay winless and bottom. Dortmund are up to fifth, but will fall to sixth should 1. FC Köln avoid defeat to Eintracht Frankfurt.