Robert Lewandowski was the hero for the home side scoring four-goals, three in the second-half after Cristiano Ronaldo had briefly levelled the contest.
"It was unbelievable, a great game by my team," Klopp told Sky Sports.
"It was so difficult to play because they are so strong and fast, and in counter attack are brilliant.
"I think the first 25-minutes were great, and then we lost a little bit of our game - not so much but enough to let them come up and they made the draw."
Klopp says that Dortmund's player were unhappy with Ronaldo's goal as they felt they should have had a penalty just moments before - but he says the players were told just to get on with the game as they had previously been on the end of a fortunate refereeing decision against Malaga in the quarter-finals.
"What we did at the break, we said maybe you think there is the wrong decision of referee' [for Real's goal].
"But the last game [against Malaga] we had the wrong decision of the referee and we were the winner, so nothing happens - the only thing that has changed was that we needed to score in Madrid - so go back to you your strong 25-minutes and take the right half with you, because the first-half the most time we played about the left wing and we wanted to play more on the right and it was much better, you saw more of the team."
Klopp hailed the attack nature of Dortmund in the second-half.
"Second half it was like Robin Hood, and arrow into the box with our attacking players - it was great to see."