The hosts were ahead after just eight minutes when Lewandowski tapped home after good work from Mario Gotze.
Dortmund were back ahead early in the second period when Lewandowski scored his second from inside the box and the Pole completed his hat-trick with a fine finish moments later.
Here, Sports Mole dissects the 90 minutes of action between the two European rivals.
On target: 7
On target: 4
Was the result fair?
There is no question that Dortmund deserved to secure a comprehensive win this evening. From the very first minute the German outfit pressed the Madrid midfield, with the likes of Mesut Ozil and Alonso simply unable to dictate proceedings. One felt that they might fall away when they were stung before the break, but they responded in the second period. They join Bundesliga rivals Bayern Munich as overwhelming favourites to book their spot at Wembley next month.
There were suggestions that Dortmund would freeze on the big occasion. Well, that was certainly not the case. It may be argued that Madrid have better individual players than the German outfit, but there is no question who the better team were this evening. From the goalkeeper to the centre-forward, each player performed their part in a terrific all-round display. The fact that they are 20 points behind Bayern in Germany tells you something about the strength of the top two in the Bundesliga.
Like their Spanish rivals Barcelona last night, Madrid seemed overawed by their German opponents. They were lucky to enter the break with an away goal, but they did have chances to add to the score in the latter stages of the first period. They simply did not start in the second half, however, with Dortmund dominating each area of the field. Jose Mourinho's side were simply unable to threaten on the counter with the compactness of the Dortmund midfield.
Sports Mole's man of the match
Robert Lewandowski: Not a difficult one. The Polish striker was in sensational form this evening as he scored four times against the current Spanish champions. His third was particularly special as he found room in a crowded box before thumping his effort into the top corner. It is easy to see why all the top clubs are in the market for the accomplished forward.
Biggest gaffe of the match
There is no question that Mats Hummels is one of the best central defenders in Europe. But he had an absolute horror show in the latter stages of the first period when he handed possession to Higuain, who squared for Ronaldo to level two minutes before the interval. That said, he responded with a fine display in the second half.
Bjorn Kuipers certainly had a tough evening in the red-hot atmosphere. The Dutchman made a big call in the latter stages of the first half when he turned down a Dortmund penalty appeal with Reus tumbling in the box. The second half was just as difficult, but he must be praised for getting the majority of the big decisions correct.
Dortmund: Dortmund are on the road this weekend as they face Fortuna Dusseldorf in the Bundesliga on Saturday.
Madrid: Mourinho's side return to domestic action on Saturday when they travel to capital rivals Atletico Madrid.