Trailing 3-0 from the first leg in Spain, Dortmund needed a minor miracle to progress to the last four of the competition and it looked as though the tie would be put beyond them when Real were awarded a penalty as Lukasz Piszczek was adjudged to have handled Fabio Coentrao's cross.
However, Angel di Maria saw his spot-kick saved by Roman Weidenfeller, and two goals from Marco Reus before half-time had Dortmund dreaming of an unlikely fightback.
They continued to create chances after the interval, Henrikh Mkhitaryan striking the post, but they could not quite force the third goal as Madrid edged through 3-2 on aggregate.
Despite exiting the competition, Klopp was proud of his players for the fight they had shown.
"Neven Subotic said to me after the game, 'I couldn't be prouder of this team'," he said.
"That's what the Champions League is about. We were superior tonight, but should have scored last week.
"Tonight's game was so good that there is absolutely no-one to blame for not qualifying.
"The atmosphere was awesome. We wanted to prove a point and we did that with our incredible performance."
Defender Mats Hummels, meanwhile, had conflicting feelings over the events of Tuesday night.
"It's a case of mixed emotions," he said. "We had our chances and came so close to clinching a miraculous win.
"We continue to prove exactly what this team is capable of against the best teams in Europe.
"The fans were exceptional. It was almost impossible to communicate with teammates they were so loud."
