Klopp’s men scored two late goals in a grandstand finish to the second-leg of their last-eight tie, to knock out the Liga side, who appeared set to spring a shock at the Westfalenstadion.
The 45-year-old is still overcome by the emotions of the night and admitted the game had been an emotional rollercoaster.
He told reporters: "It was extraordinary. I was completely broken. There were people around me who wanted to celebrate - not me. I could not, I wanted to go home because I was so broken,
"I can quite understand that the game against Malaga is still the subject of much conversation. The game felt like an eternity. It was emotional as the final whistle after the first championship win. It was something special."
Dortmund join Barcelona, Real Madrid and Bundesliga rivals Bayern Munich in the last four of Europe’s biggest club competition.
And Klopp accused the Bavarians of displaying a lack of respect for his side after FCB chief Uli Hoeness highlighted BVB as the easiest route to the final for the Bavarians.
He added: "It is understandable that they are confident.
“The 1-0 defeat to Bayern in the DFB-Pokal seems to have removed an element of respect and now we have to change this."