Marco Reus put the Bundesliga side ahead from the spot in the crucial UEFA Champions League Group F clash after Robert Lewandowski was held in the box by Federico Fernandez.
Jakub Blaszczykowski added a second on the hour mark, and Pierre-Emerick Aubameyang netted a third after Lorenzo Insigne had pulled one back for the Italians.
"We knew that this would be a spectacular game," he said. "But I didn’t expect to see so many chances and so much excitement on the pitch.
"We wasted a lot of opportunities, but Pepe Reina also made some great saves.
"We worked hard today and recorded a highly deserved win. However, Napoli was a very tough opponent and it wasn’t easy for us."
The win puts Dortmund second in the pool, ahead of Napoli on head-to-head performance with one match left to play, but defender Kevin Grosskreutz revealed Insigne's strike had left him wondering whether Dortmund had done enough to climb above the Serie A outfit.
"After Napoli scored their goal, I asked Sebastian Kehl if the result was still good enough for us," he said. "Our coach had explained everything yesterday, but I didn’t fully understand it at the time.
"After we scored to make it 3-1 I knew that was enough.
"We wanted to give our fans and ourselves something to celebrate. We deserved to win."
The German side travel to the already-eliminated Marseille for their final group fixture in two weeks' time, knowing a win will send them through to the last 16.
