Klopp’s side were on course for the quarter-finals after taking a 2-0 lead in extra time in the last-16 second leg.
Goals from Georginio Wijnaldum and Roberto Firmino left Liverpool 2-1 up on aggregate at Anfield.
But a howler from Liverpool goalkeeper Adrian opened the floodgates as Marcos Llorente struck twice before Alvaro Morata completed Atletico’s 4-2 aggregate triumph.
It was a bitter blow for Reds boss Klopp, who had seen his team dominate the first 90 minutes without taking enough chances to win the tie.
Klopp couldn’t hide his frustration at the way Atletico manager Diego Simeone set up his side in a cautious counter-attacking style for long periods.
“It’s so difficult to play a side like this. I don’t understand with the quality they have the football they play,” Klopp told BT Sport.
“They could play proper football but they stand deep and have counter-attacks.
“I realise I am a really bad loser, especially when the boys put such an effort in against world-class players on the other side who defend with two rows of four.”
The defeat ended Liverpool’s bid for a third successive Champions League final appearance.
“It doesn’t feel right. I’m searching for the right words to be honest. The 90 minutes were exceptional,” he said.
“After the second goal the legs were a bit tired. Everything that looked really natural in the first 90 minutes became a bit stiff. The crosses did not have the same quality.
“We shouldn’t have conceded these kinds of goals. It’s difficult to explain to be honest.
“Our main mistake was to not score the second goal five minutes earlier. We scored it in extra time and not in the 90 minutes.
“Everything was against us in the decisive moments. We will come again but now we are out.”
Liverpool captain Jordan Henderson insisted his team deserved to win on the balance of play, but the midfielder admitted they paid for some sloppy mistakes.
“We are really disappointed with the result because we deserved to go through,” he said.
“We are bitterly disappointed with the goals we have conceded. To concede the goals we did isn’t like us. You need to be fully focused at all times, any mistake can cost you.”
Runaway leaders Liverpool are still set to win the Premier League title despite their recent blip and Henderson said: “Tonight and tomorrow it will not feel nice but we have to react in the right way and finish the season off well.”
