Dortmund's Robert Lewandowski became the first player to score four goals in a Champions League semi-final as the Germans produced a stunning display of attacking football in the home first leg.
Klopp played down his team's performance, saying pre-tie underdogs Dortmund had done nothing more than win the first of two games. The return is at the Bernabeu on Tuesday.
"The problem is that we have played only two quarters. We still have to play the other two quarters," said Klopp, whose team are undefeated in the competition this season.
"I was happy for our very good game but I do not think we are favourites. If we lose 3-0 then we are out. To squander such a good result would be silly but nothing has been decided.
"We feel just like deserved semi-finalists. Nothing is decided and we have not too much experience at this stage.
"This is an unbelievable team, I cannot explain this in English. Robin Hood, like the arrow!"
Dortmund only reached the last four after a late two-goal burst in the quarter-final second leg against Malaga.
Klopp said Dortmund, despite losing at least one key player per season, had managed to remain a tightly-knit team which enjoyed playing together.
Mario Goetze, who created the first goal, is the next to leave after agreeing to join rivals Bayern Munich in July in a deal which angered Dortmund fans when announced on Tuesday.
"If you play football you know how to handle this situation," he said, adding fears of possible jeers and whistles from fans towards the Germany midfielder had proved unfounded.
"You cannot be successful and nobody will recognise it. It's normal to see players go. Each year we lost one brilliant player. We got better and better and better. It's no problem in the team.
"It is not easy but it's the mentality of the team to enjoy each day we are together and we will see what will be there at the end of the season."
Charismatic Klopp refused to analyse how he had stifled Real, preferring to talk about his exploits as a second division striker 25 years ago.
"I will not talk about tactics today as we need to play them again. I can say we scored great goals but that's it. I don't want to talk details because if we had a good idea that worked for us tonight I would like to keep it for me," he smirked.
Klopp was full of praise for striker Lewandowski before delivering a trademark joke.
"I scored four goals once in 1990 against Erfurt," the German coach said. "It is very high up there in my personal story and I will not forget it either."