After Bayern Munich's 4-0 hammering of Barcelona on Tuesday in the other semi-final, Borussia followed their Bavarian rivals' example, winning 4-1.
Dortmund were as clinical as Bayern as they floored Jose Mourinho's Real who narrowly avoided suffering their worst defeat in the competition - AC Milan beating them 5-0 in what was then the European Cup in the 1988/89 campaign.
"We have taken only the first step, but of course we are happy," said Lewandowski, who became the first player to score four goals in a Champions League semi-final. "I am pleased with the four goals, but now we want to reach the final."
Just as Lionel Messi had failed to shine for a subdued Barcelona at Bayern on his return from a hamstring injury, Real star Cristiano Ronaldo was fairly anonymous apart from scoring the visitors only goal.
Real coach Jose Mourinho, already a two-time winner of the trophy with Porto and Inter Milan, conceded that Dortmund had been the better side but insisted his side could pull off a remarkable comeback.
"I have learnt that anything can happen in football, it will be tough next week, but nothing is impossible and my footballers will try to show that next Tuesday," said Mourinho.
Dortmund coach Jürgen Klopp had promised his players would not be destabilised by the news on Tuesday that 20-year-old midfield star Mario Götze was to join Bayern Munich next season and his prophecy proved correct.
"We have to force ourselves to stay focused, we still need to go to Madrid, which will be no walk in the park," said Klopp referring to next Tuesday's return leg in Madrid.
The night belonged to Lewandowski who stole the limelight with the opening goal, then added a second-half hat-trick to put Dortmund on the verge of their second Champions League final having won the 1997 title.
"It will be very difficult in the return leg at home but we must remain confident and try to react," said Real captain Sergio Ramos. "When the draw for the semi-finals was made a lot of people thought this was going to be easy for us, but this is a German team we are talking about. They are a great side."
The anticipated hostile reaction from the home fans towards Götze never materialised and he helped give the hosts the perfect start.
His floated cross from the left wing found Lewandowski unmarked at the back post to stab the ball home with just eight minutes on the clock.
The goal spurred Real into life and Ronaldo forced Dortmund goalkeeper Roman Weidenfeller into a diving save on 24 minutes.
Referee Bjorn Kuipers won himself few fans among the Dortmund faithful when he refused their penalty appeal when Marco Reus was brought down in the area by Raphael Varane.
Real drew level almost immediately after that decision when Dortmund defender Mats Hummels' back pass was seized on by Gonzalo Higuain, who drew Weidenfeller and squared for Ronaldo to tap in his 11th goal in 10 European games.
The hosts upped the pace in the second half and Lewandowski grabbed his second with a clinical finish on the turn to slot home Reus' pass and beat Lopez on 50 minutes.
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He netted his third five minutes later when he latched onto Marcel Schmelzer's cross, then beat centre-back Pepe before drilling his shot home.
With Dortmund fans in delirium, he claimed his fourth of the night when Reus was fouled in the area by Xabi Alonso, after Goetze's through-ball, and there was only one candidate when Kuipers pointed to the penalty spot on 66 minutes.
Mourinho tried to breathe some life into his side with two changes as Karim Benzema came on for Higuain and Angel Di Maria took over from Luka Modric in the central midfield position.
But only Lopez's excellent diving save denied Lewandowski his fifth moments later.