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The home side almost extended their lead, but Lewandowski put his shot wide after Henrik Mkhitaryan had played him in with a defence-splitting pass.
Reina was then called into action again, tipping away Reus's effort as Jurgen Klopp's team searched for a second before half time.
Christian Maggio then failed to cut out a Mkhitaryan pass, which found its way to Lewandowski. However, the Pole's effort was comfortably saved by Reina.
A quick counter-attacking move from the home side in the early stages of the second half saw the ball fall to Reus, who had plenty of time to pick his spot, but fired his effort wide of the post.
Reina then produced an impressive save to deny Mkhitaryan as the hosts began to stretch the Italians.
However, Reina was beaten for a second time just minutes later. It all stemmed from a Napoli attack, with Roman Weidenfeller saving Gonzalo Higuain's effort. Dortmund broke and Reus slipped in a pass to the middle, where Jakub Blaszczykowski arrived to finish through the goalkeeper's legs.
The 2-0 lead lasted for little over 10 minutes, however. Lorenzo Insigne had only been on the pitch for three minutes as a replacement for Callejon, when he chipped the ball over the on-rushing Weidenfeller.
Unperturbed by that setback, Dortmund pushed for a third goal and Reina needed to be in top form to keep out Pierre-Emerick Aubameyang.
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However, Aubameyang found the net minutes later. Pablo Armero gave the ball away to Lewandowski and the Pole played in Aubameyang. The striker collected the ball and clipped it over Reina for 3-1.