Daniel Svensson and Fabio Silva got on the scoresheet for the visitors, who also benefitted from an own goal from HEBC defender Janek Wrede.
Officially the hosts, HEBC moved the match from their 2,500-seat home to the 57,000-strong Volkspark Stadion, home of Bundesliga side Hamburg.
Despite surrendering home advantage with much of the crowd bathed in Dortmund yellow, the official attendance was 45,917, a German Cup record for a match involving a fifth-tier team, while much of the revenue will flow to the amateur side.
Dortmund coach Niko Kovac made seven changes, although deadline-day newcomer Ethan Nwaneri arrived too late on his season-long loan from Arsenal to take part.
HEBC held the 2024 Champions League finalists at bay for the opening half-hour but Dortmund's pressure soon told, with the hosts conceding an own goal in the 33rd minute before Svensson netted three minutes later.
Teenage forward Inacio then bagged a brace before half-time to put any hopes of a shock upset well beyond doubt.
Former Wolves forward Silva added a fifth just before the hour mark to add gloss to Dortmund's victory.
On Wednesday, holders Bayern Munich play at second-tier Osnabruck in their German Cup opener.