Vinicius Junior scored either side of Marco Asensio's goal as LaLiga champions Madrid secured a 3-1 victory in the opening leg of the blockbuster tie, ahead of next week's return fixture at Anfield.
Mohamed Salah scored for Premier League holders Liverpool early in the second half, but the visiting Reds were the masters of their own downfall from a defensive perspective, especially in the opening 45 minutes.
The centre of much media discussion following his omission from Gareth Southgate's latest England squad, Liverpool star Trent Alexander-Arnold endured a first half to forget in the Spanish capital as he failed to track Vinicius for Madrid's opener – created by an exquisite Toni Kroos pass – before the full-back headed straight into Asensio's path nine minutes later.
Liverpool's defence was exposed again when Vinicius found space in the box to drill in his second and restore Madrid's two-goal cushion – goalkeeper Alisson also at fault after getting a strong hand to the ball.
Klopp's men failed to have a single attempt in the first half, the first time Liverpool have not managed a shot in the opening period of a Champions League match since 2014, and the German manager was left far from satisfied.
"If you want to go to the semi-finals you have to earn the right to do so," Klopp told BT Sport.
"We didn't do that, especially in the first half, so the only good thing really that I can say about the game, apart from that we scored a goal, is that it's only the first half of the leg.
"We just didn't play good enough football to cause Real Madrid more problems. I really thought it would have been possible. We made it too easy for them, that's the main thing."
Asked if he was concerned by more dubious defending, Klopp instead cited Liverpool's poor use of the ball as their main issue, though the statistics contradicted his argument slightly – the Reds completing 85.8 per cent of their 605 passes, in contrast to Madrid players successfully finding team-mates with 82.1 per cent of their 507 passes.
"The goals in the end, are quick thinking from Real Madrid, but these are mistakes that can happen," he said.
"I wasn't happy with the way we played football in possession. He situations when we pressed really well, we'd win that ball but the second pass was away again, that was the problem.
"We cannot defend against the speed of Real Madrid if we give them the ball in the completely wrong moment, so if you lose the ball it's not a problem in the right moment if you counter-press then you get the ball back.
"But we lost the ball in the wrong moments, passing it to their feet and then what they want to do is to switch the play, accelerate, with really quick and world class players in a one-on-one situation, it's not cool, not easy for whoever is in the challenge for us. We have to be better defensively but it's not about the last line, it's about what we do in front of that.
Across European competitions, no side has beaten Liverpool more often than Madrid (four, level with Benfica), while Los Blancos have progressed from 15 of their last 16 Champions League knockout matches after winning the opening leg, only failing to do so against Dutch giants Ajax in 2018-19.
It is a daunting challenge facing Liverpool, who have not won at Anfield since December, where they have lost their last six home games – the club's 2-0 victory in the home leg of their last-16 tie with RB Leipzig having taken place in Budapest due to coronavirus travel restrictions between Germany and Britain.
"Intensity with the ball, especially," Klopp replied when asked what he was looking for in the return fixture.
"We have to create, play better football. We didn't deserve a lot more tonight, one goal in the second half was okay, it gives us a lifeline."