A dramatic second-half spell saw four goals in 14 minutes as Oleg Shatov and Hulk, with a penalty, scored for Zenit and Lewandowski quickly replied each time for the visitors.
Zenit, who are trying to reach the quarter-finals for the first time, were left with an almost impossible task of having to score at least three goals in the return on March 19.
Zenit were quickly in trouble as Reus burst into the area and, although he stumbled, the ball ran kindly to Mkhitaryan and the Armenian fired the ball past Yuri Lodygin.
The Zenit goalkeeper had to pick the ball out of his net again 70 seconds later after Mkhitaryan burst down the right and found Kevin Grosskreutz who laid the ball off for Reus to score with a clinical first-time finish.
Zenit were penned back in their own half and their problems mounted when midfielder Andrey Arshavin went off injured and was replaced by Anatoliy Tymoshchuk.
Dortmund began the second half on the attack as their Poland striker Lewandowski had a powerful shot turned around the post by Lodygin.
The Russians reduced the deficit in the 57th minute with a scrambled goal when Oleg Shatov fired home from the rebound after Jose Salomon Rondon had one effort headed away and then hit the post.
Lewandowski restored Dortmund's two-goal advantage four minutes later after combining with compatriot Lukasz Piszczek in a counter-attack down the right.
Hulk pulled another back for the hosts with a controversial penalty in the 69th minute, awarded after Viktor Fayzulin dribbled into the area and collided with Piszczek.
But Zenit gifted Dortmund another goal when Tymoshchuk was dispossessed in his own half by Reus, who slipped the ball through for Lewandowski to score with a low, angled shot.
(Reporting by Brian Homewood; editing by Ken Ferris)
