Liverpool's Champions League hopes ended at the hands of Real Madrid for a third consecutive season as they failed to overcome their first-leg Anfield demolition.
The Reds, who lost the first leg 5-2, lined up with four forwards at the Bernabeu as they looked to add another famous comeback to their fabled European history.
But that rarely looked on the cards and legendary Real striker Karim Benzema put the tie beyond doubt when he tucked in from Vinicius Jr's cutback late on.
Jurgen Klopp's side would have lost this game by more if not for Alisson.
The Reds goalkeeper denied Vinicius and Eduardo Camavinga with magnificent first-half stops.
If Darwin Nunez had taken a chance in the sixth minute, then it could have been a different story.
Liverpool players walked off to 'You'll Never Walk Alone' on the Bernabeu speakers as they reflected on their biggest ever aggregate Champions League defeat.
The Reds were beaten finalists against Real last summer but now face a battle to even be in the tournament next season, sitting six points behind the Champions League places.
One downside for Carlo Ancelotti's Real was Benzema hurting himself while scoring and having to be replaced, four days before their must-win El Clasico with Barcelona.
No miracle of Madrid
This was the 92nd time a team have lost by three or more goals in the first leg at home in a European Cup tie. All 92 have been eliminated.
Data analysts Gracenote gave Liverpool a 3.92% probability of progression and Jurgen Klopp talked about a "1% chance" beforehand.
That was the challenge they faced after Madrid put on an exhibition of taking their chances at Anfield.
But still, this is Liverpool and this is the Champions League.
In 2005 they came from 3-0 down to beat Ancelotti's Milan in the Champions League final. Three years ago they lost 3-0 to Barcelona and won the second leg 4-0, although crucially that was at home.
Liverpool gave it a go - they had to - with forwards Diogo Jota, Nunez, Mohamed Salah and Cody Gakpo in their line-up.
Had Nunez scored after six minutes instead of shooting at Thibaut Courtois, who knows what might have happened. The Uruguayan forced another two saves from the goalkeeper, most notably with a fine curling effort after cutting in from the left wing.
But really they were lucky to only lose this leg 1-0. There was no gung-ho, 'heavy metal', blockbuster attempt to go and win it.
They did not even have a shot between James Milner's blocked effort in the 37th minute and Harvey Elliott's attempt on target after 82 minutes.
It could have got slightly worse in injury time when referee Felix Zwayer went to the monitor to see if Konstantinos Tsimikas had handled the ball, but the decision was no penalty to Real.
So just like in last year's final in Paris, and the previous season's quarter-final, Real were their Liverpool's conquerors.
The Reds will undoubtedly finish this season trophy-less and it will be a huge blow to their future plans if they cannot overtake Newcastle and Tottenham to finish fourth in the Premier League come the end of the season.
Champions League kings march on
Ancelotti said beforehand that psychologically "it's tougher for us than Liverpool" because the Reds knew they had to go "all out from the first minute".
But it was Real who had the bulk of the chances. They had 17 shots here, compared to the nine efforts that yielded five goals on Merseyside.
Alisson somehow saved Vinicius' close-range volley with an outstretched arm. Then he tipped Camavinga's deflected, looping long-range strike on to the crossbar.
Luka Modric drilled a shot just over from outside the box. After the break Alisson denied Federico Valverde and Benzema in quick succession.
Valverde was unmarked when he headed another chance over.
Eventually the pressure told when Benzema slotted in from Vinicius' ball as they got in behind the Reds' defence. That was his seventh Champions League goal against the Reds, a record for any player.
Ancelotti and Real will hope the problem that saw him go off is not too bad. On Sunday they go to the Nou Camp needing a win to keep their La Liga title hopes alive, as they trail leaders Barcelona by nine points.
But for now the 14-time European champions can deservedly celebrate reaching a 19th Champions League quarter-final.
Player of the match
ModricLuka Modric
Real Madrid
Squad number10Player nameModric
Squad number20Player nameVinícius Júnior
Squad number9Player nameBenzema
Squad number12Player nameCamavinga
Squad number8Player nameKroos
Squad number15Player nameValverde
Squad number18Player nameTchouaméni
Squad number22Player nameRüdiger
Squad number19Player nameDani Ceballos
Squad number17Player nameLucas Vázquez
Squad number11Player nameAsensio
Squad number3Player nameÉder Militão
Squad number6Player nameNacho
Squad number1Player nameCourtois
Squad number21Player nameRodrygo
Squad number2Player nameCarvajal
Line-ups
Real Madrid
Formation 4-3-3
1Courtois
2Carvajal3Militão22Rüdiger6Nacho
10Modric12Camavinga8Kroos
15Valverde9Benzema20Vinícius Júnior
1Courtois2CarvajalSubstituted forVázquezat 86'minutes3Militão22Rüdiger6Nacho10ModricSubstituted forCeballosat 82'minutes12Camavinga8KroosSubstituted forTchouaméniat 84'minutes15Valverde9BenzemaSubstituted forRodrygoat 82'minutes20Vinícius JúniorSubstituted forAsensioat 84'minutesSubstitutes
5Vallejo7E Hazard11Asensio13Lunin16Odriozola17Vázquez18Tchouaméni19Ceballos21Rodrygo23Mendy24Mariano39RodríguezLiverpool
Formation 4-3-3
1Alisson
66Alexander-Arnold5Konaté4van Dijk26Robertson
18Gakpo3Fabinho7Milner
11Salah20Jota27Núñez
1Alisson66Alexander-Arnold5Konaté4van Dijk26RobertsonSubstituted forTsimikasat 90+1'minutesBooked at 90mins18GakpoSubstituted forCarvalhoat 90+1'minutes3Fabinho7MilnerSubstituted forOxlade-Chamberlainat 73'minutes11Salah20JotaSubstituted forElliottat 57'minutes27NúñezSubstituted forFirminoat 57'minutesSubstitutes
2Gomez8Keïta9Firmino13Adrián15Oxlade-Chamberlain17Jones19Elliott21Tsimikas28Carvalho32Matip46R Williams62KelleherReferee:Felix ZwayerAttendance:63,127Live Text
Match ends, Real Madrid 1, Liverpool 0.
Second Half ends, Real Madrid 1, Liverpool 0.
Konstantinos Tsimikas (Liverpool) is shown the yellow card for a bad foul.
Dani Ceballos (Real Madrid) wins a free kick on the right wing.
Foul by Konstantinos Tsimikas (Liverpool).
VAR Decision: No Penalty Real Madrid.
Attempt blocked. Rodrygo (Real Madrid) right footed shot from the centre of the box is blocked. Assisted by Lucas Vázquez.
Substitution, Liverpool. Fábio Carvalho replaces Cody Gakpo.
Substitution, Liverpool. Konstantinos Tsimikas replaces Andrew Robertson.
Nacho (Real Madrid) wins a free kick on the left wing.
Foul by Harvey Elliott (Liverpool).
Substitution, Real Madrid. Lucas Vázquez replaces Daniel Carvajal.
Corner, Liverpool. Conceded by Antonio Rüdiger.
Foul by Éder Militão (Real Madrid).
Cody Gakpo (Liverpool) wins a free kick in the attacking half.
Substitution, Real Madrid. Aurélien Tchouaméni replaces Toni Kroos.
Substitution, Real Madrid. Marco Asensio replaces Vinícius Júnior.
Foul by Nacho (Real Madrid).
Harvey Elliott (Liverpool) wins a free kick in the attacking half.
Attempt saved. Harvey Elliott (Liverpool) left footed shot from outside the box is saved in the top left corner. Assisted by Fabinho.