Lewandowski became the first player to score four goals in a Champions League semi-final.
Dortmund were as clinical as Bayern as they floored Jose Mourinho's squad. Real, in fact, narrowly avoided suffering their worst defeat in the competition. They had suffered a 5-0 loss to AC Milan in 1980s.
Real star Cristiano Ronaldo was fairly ineffective apart from scoring the visitors only goal. Coach Mourinho 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 Jurgen Klopp had promised his players would not be destabilised by the news on Tuesday that 20-year-old midfield star Mario Goetze was to join Bayern Munich next season and his prophecy proved correct.
Lewandowski stole the limelight with a goal first half and 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.