The holders looked to have ended Atletico's dogged resistance when Roberto Firmino put them ahead in the last-16 tie after Georginio Wijnaldum's first-half header forced the second leg into an extra 30 minutes
Atletico Madrid manager Diego Simeone defended his side's approach after dumping Liverpool out of the Champions League with a 3-2 victory after extra time at Anfield to progress to the quarter-finals, winning 4-2 on aggregate.
The holders looked to have ended Atletico's dogged resistance when Roberto Firmino put them ahead in the last-16 tie after Georginio Wijnaldum's first-half header forced the second leg into an extra 30 minutes.
However, Liverpool were made to rue a host of missed chances to kill the tie off in 90 minutes as Llorente twice curled home from outside the box before Alvaro Morata twisted the knife by breaking away to score in added time of extra time.
When you knock out the reigning European champions pic.twitter.com/X0tE5daz52— B/R Football (@brfootball) March 12, 2020
Defeat ends Jurgen Klopp's proud record of never losing a two-legged European tie as Liverpool manager and he took a swipe of Atletico's defensive tactics.
"The way they play, I don't get it," said Klopp. "They could play proper football but they stand deep and have counter-attacks."
It remains to be seen if and when Atletico will play in the last eight due to the chaos caused across the continent by the new coronavirus.
But Simeone basked in another famous European night for the side he has twice led to the final in his eight years in charge.
"We play to win, with the arms that we have," said Simeone. "Respecting our identity, the characteristics of our players and exploiting the defects of our rivals."
A packed Anfield played its part in trying to roar Liverpool onto another Champions League conquest as, unlike in numerous countries across Europe, no restrictions have so far been put on supporters attending games in the United Kingdom.
Diego Simeone defended his side's approach after dumping Liverpool out of the Champions League with a 3-2 victory after extra time.— SuperSport 🏆 (@SuperSportTV) March 12, 2020
"We play to win. Respecting our identity, the characteristics of our players and exploiting the defects of our rivals."https://t.co/4M9eQQi7MZ
But it was the nearly 3,000 travelling fans in attendance, despite the fact games in Spain will be played behind closed doors for the next two weeks, that celebrated a stoic rearguard display from their side.
The European champions did not manage a single shot on target in a 1-0 defeat in Madrid three weeks ago, but Jan Oblak was quickly put to work.
But Oblak was helpless as Liverpool levelled the tie just before half-time through Wijnaldum's powerful downward header.
The goal did not change Atletico's approach as they played a risky game and relied on Oblak to keep them in the game.
The Slovenian saved from Mohamed Salah, Alex Oxlade-Chamberlain and Firmino, while Andy Robertson hit the bar as Liverpool missed a number of chances that ultimately came back to cost them.
"I have no doubt Oblak is the best goalkeeper in the world," added Simeone.
1 - Atlético de Madrid yesterday v Liverpool :— OptaJean (@OptaJean) March 12, 2020
3 shots in extra time ☄️
3 goals ⚽️
First team to win a Champions League game after trailing by two goals during the extra time
Cholo. @Atleti @OptaJose pic.twitter.com/QnHAW8KYeW
Despite Liverpool being on the brink of a first Premier League title in 30 years, their number nine Firmino had not scored at home all season until four minutes into extra-time.
The Brazilian's header from Wijnaldum's cross came back off the post and perfectly back into his path to slot home with Oblak grounded.
"Everybody who saw the game tonight knows it could have been different," said Klopp. "I loved our first 90 minutes, but we scored the second goal too late."
However, no sooner had Liverpool gone in front for the first time in the tie than Atletico hit back with a sucker-punch.
The visitors had barely tested stand-in Liverpool goalkeeper Adrian, but the loss of Alisson Becker to a hip injury eventually proved Liverpool's downfall.
Adrian fired a simple clearance straight to Joao Felix, who fed Llorente and he found the bottom corner.
Llorente goal Liverpool vs atletico madrid champions league new 2-1 pic.twitter.com/bNKub2Xaj0— Fightgame (@futurefights360) March 11, 2020
"The pass was not helpful. Adrian is a super player, I love the boy, but in this moment, it is the wrong decision," added Klopp.
Llorente had joined Atletico from bitter rivals Real Madrid in a controversial move in June.
But the man whose father and great uncle, Francisco Gento, also played for Real, made himself a Rojiblanco hero with a second that secured Atletico's place in the last eight with another precise low finish in added time at the end of the first half of extra time.
Another former Real player Morata then broke away and finished calmly to inflict Liverpool's first Champions League home defeat since 2014.