The visitors had a chance to add to their three-goal advantage after the first leg when they were awarded a penalty for a handball, but Angel di Maria's spot kick lacked power, allowing Roman Weidenfeller to make the save.
The missed penalty appeared to give the Bundesliga side a boost, as Reus quickly found his way into the box at the other end of the pitch, but Henrikh Mkhitaryan could only turn his delivery wide.
Dortmund continued to look the more likely to score, and Reus was able to capitalise when Pepe tried to head the ball back to keeper Iker Casillas, but the midfielder reached it first, rounding the Real goalie before slotting the ball in.
The hosts almost added a second when Reus delivered a free kick for Mats Hummels, but his header just cleared the crossbar.
However, Reus grabbed his second of the evening shortly before the break, smashing in the rebound after Robert Lewandowski struck the post.