Germany international Marco Reus converted a 10th minute penalty and Jakub Blaszczykowski added another goal on the hour before Lorenzo Isigne's 71st minute strike set up what looked like a nervous finale to the Group F match.
However, Dortmund substitute Pierre-Emerick Aubameyang made sure of the three points with his first Champions League goal seven minutes later to lift Dortmund into second place ahead of Napoli with both teams on nine points.
Dortmund, who got their first win after three straight defeats in all competitions, including a 3-0 Bundesliga loss to rivals Bayern Munich, visit Olympique Marseille in their last group match next month.
Napoli were in no mood to play for a draw, with coach Rafa Benitez fielding an offensive formation with four players supporting lone striker Gonzalo Higuain.
Dortmund, with their backs to the wall and missing their entire first choice defence through injury, got off to a perfect start when Reus converted an early spot kick awarded for handball by Raul Albiol to put the Germans in the lead.
With the goal settling Dortmund's nerves, Robert Lewandowski should have added another five minutes later but his shot, with only goalkeeper Pepe Reina to beat, sailed narrowly wide.
Reus then tried to beat the Spaniard with a clever free kick but Reina palmed it wide.
Napoli, eager to get their season back on track after two straight league defeats, briefly took the initiative and Jose Callejon came agonisingly close to an equaliser after a fine run down the right when his shot hit the base of the far post.
But it was Reina who was the busiest stopper, twice more denying Poland striker Lewandowski as Dortmund desperately looked for a second goal.
Chances continued to fall for Dortmund, who upped the tempo in the second half but again were repeatedly denied by Reina, who is on loan from Liverpool.
He could do nothing, however, when Reus found Blaszczykowski in the box and the Pole slotted in for a two-goal cushion on the hour following a dazzling quick break.
Insigne cut the deficit after a fine pass from Higuain but Aubameyang's 78th minute goal killed off any Italian hopes.
(Reporting by Karolos Grohmann; editing by Ken Ferris)
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