Gareth Bale got things rolling with the opening goal inside three minutes and Isco added a second before Cristiano Ronaldo continued his stunning form in the competition with Madrid's third just before the hour.
That saw Ronaldo equal the Champions League record of 14 goals in a season and made Madrid firm favourites to progress through and avenge last season's semi-final exit at the hands of Dortmund.
Carlo Ancelotti's men could have added even more goals but found Dortmund goalkeeper Roman Weidenfeller in fine form, while the under-strength visitors wasted numerous chances to score the away goal which could have put a different complexion on things for the return leg at the Westfalenstadion.
Klopp could name in his side only four of the side that also started the famous 4-1 win over Real in last season, but the coach refused to blame absences for the defeat.
"Yes it is a difference, but the goals we conceded tonight have nothing to do with that," he said.
"We were not compact enough. The first goal was too easy, we made things too easy for them.
"In the second half we had so many chances and we didn't take one of them, but I know this team can do so much better and that is what we have to think about."
Goalscorer Isco was delighted to establish such an advantage ahead of next week's second leg, but the young Spaniard insists his team-mates will not yet take qualification for granted, especially after their experience at the Westfalenstadion last year.
"It's a magnificent result," Isco told Canal Plus.
"We are very happy to get the third goal and not concede. It is a huge step towards the semi-finals."
"But no one imagined Madrid would lose 4-1 over there. In football you never know."