Dortmund lost its last three matches, against Arsenal and two in the Bundesliga, and was missing its entire regular back four because of injuries. Sven Bender, normally a midfielder, was pulled into defense and played through despite sustaining a broken nose in the first half.
Needing a win to remain in contention, Klopp's team got onto the scoreboard fast.
Following a corner, Federico Fernandez hugged Robert Lewandowski until finally pulling the striker down to give away a penalty.
Reus stepped up to send Pepe Reina the wrong way in the 10th minute.
Dortmund controlled the game and did most of the attacking but could not convert its chances. Napoli came close to equalizing when Jose Callejon cut inside past Bender and beat Roman Weidenfeller with a low shot to the far corner but the ball bounced off the post.
With both goalkeepers enjoying a good night, Reina denied Henrikh Mkhitaryan, who had broken through past three defenders. Reina stuck out his hand to block the shot.
At the other end, Gonzalo Higuain was clean after a through ball but Weidenfeller reduced the angle and stopped the shot.
Dortmund replied immediately with a typically swift counterattack. Reus sent a perfect pass into the middle where Blaszczykowski slotted the ball through Reina's legs with a first-time shot in the 60th minute.
However, Napoli got back into the match in the 71st. Sebastian Kehl gave the ball away and Higuain played it through to substitute Insigne, who beat Weidenfeller with a shot into the far corner.
Another substitute secured Dortmund's win.
Aubameyang wasted one opportunity when Reina stuck out his foot to block the point-blank attempt but in the 78th the forward made no mistake.
Pablo Armero lost possession, Robert Lewandowski slipped the ball to Aubameyang to the right and he chipped it over Reina.
"We had many chances but Reina was very strong," Klopp said.
