Last year's beaten finalists produced a stirring performance to give the Spanish giants a scare in front of an ear-splitting home crowd.
A missed penalty from Angel Di Maria spurred the hosts on and Marco Reus netted twice as Dortmund reduced their arrears to one before the half-time break.
Head coach Klopp hailed the spirit within the Dortmund camp and believes plenty can be learned from his side's thrilling comeback attempt.
"When I high-fived our injured player Neven Subotic after the game, he told me how proud he was to be a part of this team and this is exactly how I see it as well," he told UEFA's official website.
"You simply had to produce a video of this game and show it to all the teams that have ever lost a first leg 3-0."
Klopp acknowledged the desire of his side on home soil and believes nobody can be blamed after an "extraordinary" performance.
"The lads wanted to go through, we were unable to show that in Madrid, but today we were capable of showing that 100 per cent," he added.
"We were clearly the better side today and we really should have scored more goals, but because the game and our performance was so extraordinarily good I really will not make accusations to anyone today.
"The atmosphere was extraordinary, but we were capable of lighting an early beacon of hope to show that something was still possible tonight - it was a simply extraordinary night."
Dortmund centre-back Mats Hummels concurred, claiming the noise within the Westfalenstadion was overwhelming at times.
"The reaction of our fans was incredible," he said.
"You could hardly give instructions to your team-mates in the first half because it was so loud."