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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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."
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.
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.
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.
