But while Dortmund dominated, it was unable to add to its advantage and was made to pay as Real struck back with just two minutes of the half remaining.
With the home players still appealing for a penalty for what they considered to be a foul on Reus, the visiting team took advantage in devastating fashion.
A long ball was misjudged by Mats Hummels and Gonzalo Higuain ran clear to square for Cristiano Ronaldo to roll the ball home.
Ronaldo's strike, his sixth in six consecutive Champions League games, appeared to have left Dortmund in a daze.
But whatever Klopp said to his side at the break worked a treat as they produced one of the most enthralling second half displays in recent times.
It was Lewandowski who led the charge, firing home his second within five minutes of the restart after taking Reus' deflected effort in his path.
His third followed soon after, and when it came, it was worth waiting for.
Marcel Schmelzer's cross-cum-shot found the forward inside the penalty area and the striker produced an exquisite turn before sending an unstoppable effort into the roof of the net.
Dortmund were rampant and the hugely impressive Gundogan almost added a fourth when his fierce effort was brilliantly tipped away by Lopez in the Real goal.
That fourth goal was not far away, however, and when Xavi Alonso fouled Reus inside the penalty area, Lewandowski stood up to power home from 12-yards.
Not since the legendary Hungarian striker Ferenc Puskas in 1960 had anybody scored four times in a European Cup semfinal or final.
That tally could have been greater but Lopez produced another stunning save to push the Pole's fierce effort over the crossbar.
Real did rally late on with Ronaldo and Raphael Varane both going close, but Jose Mourinho's men must now hope for an incredible turnaround next Tuesday.
Only once has a team emerged victorious after conceding a three goal deficit in a Champions League knockout stage.
On that occasion, Deportivo La Coruna came back from a 4-1 defeat at AC Milan to win 5-4 on aggregate courtesy of a 4-0 home win.
"I think Dortmund were the best team by far," Real boss Mourinho told Sky Sports.
"They won every individual battle on the pitch, they were stronger physically and mentally so I think they deserve it. How did it go from 1-1 to 4-1? I don't know.
"Everything happened in a short space of time. We lost easy possession and we couldn't cope with their counter attack and their transition.
"We know everything about Robert Lewandowski, every detail possible and we lost him in three goals. It's very disappointing.
"When we lose and we don't deserve to lose, with me, it's a drama, when I do deserve to lose I accept it. Of course the boy deserves credit for what he did but we gave him every support for man of the match."
A 3-0 win would take Real through to the final and keep alive its hopes of winning "La Decima" -- the club's tenth trophy in the competition.
But the task looks nearly impossible following its abject showing in Dortmund.
"It will be very difficult in the return leg at home but we must remain confident and try to react," Real captain Sergio Ramos told reporters.
"When the draw for the semifinals 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."