The Poland striker gave Jurgen Klopp's men an early lead and, after Cristiano Ronaldo equalised late in the first half for Real Madrid, Lewandowski struck three times more in the second half to secure a 4-1 win.
After 2012 finalist Bayern's 4-0 hammering of Barcelona the previous night, it was a further indication ahead of next week's return legs in Spain that the balance of power in European football may be shifting.
After a build-up dominated by the announcement that Mario Gotze will depart Dortmund for Bayern in the northern summer, his current side made a blistering start.
Raphael Varane cleared Marcel Schmelzer's early cross but Dortmund soon had a clear opportunity.
Marco Reus, of whom even more will be expected once Gotze departs, isolated Varane and surged through the visiting defence. His shot was parried by Diego Lopez and Lewandowski, reluctant to over-stretch, failed to tuck away the loose ball.
The goal arrived in the eighth minute, though, Gotze swinging in a superb cross from the left for Lewandowski, this time reaching out a long right leg after losing his marker Pepe, to sidefoot home.
Real began to establish itself in the game as the tempo settled down, and a 30-yard free-kick provided its first sight of goal. Ronaldo's dipping effort was beaten away by goalkeeper Roman Weidenfeller.
Another wonderful run by Reus was broken up on the edge of the box and Fabio Coentrao's interception then halted a lightning break.
Jakub Blaszczykowski caused problems with a surge into the Real penalty area, past Coentrao, but Gonzalo Higuain had tracked him all the way back and the striker made a superb challenge as the last man.
And having excelled defensively, Higuain then played his part at the other end as he seized on Mats Hummels' poor backpass and squared for Ronaldo to slot the ball home.
That left the scores level at the break but Lewandowski restored his side's lead five minutes after the restart.
With Real appealing for offside as Reus' pass found the Pole he kept his head, turned quickly on the ball and poked low past Lopez.
And it was soon 3-1 as Schmelzer's mishit shot deflected to Lewandowski, who brilliantly dragged the ball back away from Pepe and fired it into the top corner.
Lopez had to be at his best to keep Real in the game when Ilkay Gundogan drove through the defence and punched a rising effort at goal, which the keeper tipped over the bar.
And dominant Dortmund was then awarded a penalty for Xabi Alonso's blatant shove in the back of Reus. Lewandowski slammed the penalty straight down the middle to make it 4-1.
The striker put a slight blot on his copybook with a booking for blocking Sergio Ramos but soon threatened a fifth goal with a 25-yard curler pushed over by Lopez.
A tackle between Hummels and Ronaldo punctured the ball and meant a replacement was needed, and Weidenfeller was injured in another challenge with the Portugal international.
Four added minutes passed with Varane's shot, which flashed narrowly wide, the only sight of goal.