The Spaniards had no answer for the goal-hungry Pole, who was a one-man wrecking ball as Jurgen Klopp's charges inflicted a painful defeat on the visitors
The Poland star was on target for all four of his side's goals as they overcame the Blancos, whose only consolation was an away goal courtesy of Cristiano Ronaldo that gives them the faintest of hopes for the return.
Looking to follow the example of compatriots Bayern Munich after the Bavarians destroyed Barcelona the previous evening, Dortmund came out in a bullish fashion and almost took the lead in the first few minutes.
Marco Reus picked up the ball on the halfway line and embarked on a brilliant run, weaving past several Madrid markers before squaring up to Diego Lopez in goal. The keeper did well to parry the shot, and Lewandowski found himself on the wrong side of the rebound and could not convert.
If the Poland international was irritated with himself at letting the chance slip away, he made instant amends. A fluid move from the home side ended at the feet of Mario Gotze, and the young forward looped a cross into the area. Lewandowski reacted before anyone in a white shirt, muscling his way between the Merengues defence and diving in to steer home to put his team in the ascendancy after just eight minutes.
The goal was no less than BVB deserved for an all-action start to the clash, as they harried their rivals across the park and gave them barely a second to breathe.
As the half progressed, Madrid started to gain a foothold and enjoy more possession, although clear chances were at a premium. It was from dead balls that they looked most dangerous. Several deep crosses required strong defensive action from the Germans, while Ronaldo went closest to troubling the scorers with a long-range free kick that was parried away by Roman Weidenfeller.
Just as the Spanish side looked certain to enter the break one down, two big incidents changed the course of the half. A clumsy challenge from Rafael Varane on Reus as the forward escaped into the box provoked desperate calls for a penalty, shouts that fell on death ears as the referee refused to oblige. Almost without pause, Madrid took their chance to strike.
A direct counter looked to be straightforward for Mats Hummels, but an attempted back-pass from the defender was well short and seized upon by Gonzalo Higuain. The Argentine could have gone alone behind the backline, but instead chose to square perfectly for Ronaldo who stroked into the empty net to level the game minutes before half-time.
Dortmund could have panicked after seeing their lead evaporate, but instead they opted to start the second period in exactly the same fashion as the first. With just four minutes played after the break, Reus picked up a header on the edge of the box and threaded through to Lewandowski. Madrid's defence screamed offside, but it was inattention from Varane that led to the striker being left all alone, and he made no mistake in punishing the error as he slammed past Lopez for his and BVB's second.
The Bundesliga's top scorer was in no mood to slow down. With Madrid still reeling from that early blow, Marcel Schmelzer picked up a loose ball on the edge of the area and hit with venom towards goal. The shot took a deflection and fell perfectly for Lewandowski, who executed a marvellous turn and shot into the roof of the net to seal his hat-trick, put his team two goals clear, and in control of the tie.
The consolation for Madrid is that it could have been even worse. Ilkay Gundogan left his marker standing with a silky turn and shaped towards goal, launching a stinging effort from the left. The ball looked destined to end in the top corner if not for the intervention of Lopez, who pulled off a stunning fingertip save to avoid a fourth and further humiliation for Jose Mourinho's charges. The goalkeeper was let down, however, by some boneheaded play from those in front of him.
Madrid were resigning themselves to chasing a two-goal deficit in the Bernabeu, albeit with an away strike, when an idiotic shove from Xabi Alonso on Reus led to a penalty. Who else was going to step up: Lewandowski prepared calmly and blasted past Lopez for his fourth, to cap an exceptional display for both the striker and Dortmund, who held on to record an impressive win over the Spaniards.
The Merengue were left bruised after a convincing defeat, but will still hope to turn round the fixture thanks to Ronaldo's away goal. The second leg will be played in the Santiago Bernabeu, on April 30.