Dortmund's dazzling performance comes only a day after Bayern Munich crushed Barcelona 4-0 in the other semi-final first leg to raise the prospect of an all-German match-up in the final at Wembley in May.
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Poland international Lewandowski, who looks set to leave after refusing to extend his contract past 2014, became the first person to score four goals in a Champions League semi-final, opening his account after eight minutes.
The 24-year-old added three more in the second half, including an emphatic penalty, leaving the nine-time European champions badly wounded.
Cristiano Ronaldo had equalised just before the break for the Spaniards, who face a monumental task against still undefeated Dortmund in Tuesday's home second leg.
"We lacked commitment, starting with myself and including the 10 others who were on the pitch," Real defender Sergio Ramos told Canal Plus.
"We have to keep faith that we can turn it around in the return leg, there are 90 minutes left. We will come out fighting from the first minute because it's a competition that means a lot to us."
Dortmund, who had beaten Real at home in the group stage, had hoped a surprise July transfer of Mario Gotze to Bayern Munich announced on Tuesday would not distract them.
Coach Jurgen Klopp had appealed for a great footballing occasion and the 62,000 Dortmund fans, angered by the 20-year-old's decision to join their arch-rivals in the south, showed restraint when he came onto the pitch.
He soon paid them back, curling a superb cross into the box for Lewandowski, who had come close a minute earlier, to tap in after holding off Real's Pepe for his seventh goal of the Champions League campaign.
ONE-SIDED TRAFFIC
It took Real more than 20 minutes to curb their opponents' attacking game and Ronaldo tested goalkeeper Roman Weidenfeller with a long-range stinging free kick.
But they needed a Dortmund mistake to level in the 43rd, seconds after the Germans had demanded a penalty for a challenge on Marco Reus.
Central defender Mats Hummels miscued his back pass and Gonzalo Higuain intercepted it and fed unmarked Ronaldo in the box who levelled with his 12th goal of the competition.
Ronaldo's goal meant he equalled a record of scoring in six straight Champions League games and joins Thierry Henry in fourth place of the Champions League all-time scorers list with his 50th goal.
But Real's joy was shortlived when Lewandowski struck twice in five minutes after the break as the one-sided traffic resumed.
The Pole was picked out by Reus in the 50th minute to score from close range after evading the offside trap and then controlled a Marcel Schmelzer cross to drill in from about 10 metres after some superb skill.
He completed a brilliant individual and team performance when he converted a 66th m
