The Reds improved from their 5-2 thrashing in the first leg at Anfield but were comfortably second best to Los Blancos at the Santiago Bernabeu. Alisson was by far the visitors' most impressive player on the night, making a string of brilliant saves, while Liverpool struggled to create any clear-cut opportunities for their attacking line-up to capitalise on.
Real Madrid scored a deserved goal to extinguish any hopes of a late Liverpool miracle as Karim Benzema tapped into an open net with just over 10 minutes remaining.
A European exit at the last-16 stage means Jurgen Klopp's men will have devote all their attention to the Premier League top-four battle. Real Madrid, meanwhile, underlined their credentials to defend their Champions League crown with a composed, intelligent performance in front of their own fans.
Here are five talking points from the Spanish capital.
Was Liverpool's aggressive set-up due to having nothing to lose or a lack of available midfielders?
Whatever Jurgen Klopp's reasoning, the Reds lined up with four recognised forwards at the Santiago Bernabeu - a strategy few visiting teams have employed at the palatial stadium in recent years. Diogo Jota led the line flanked by Mohamed Salah and Darwin Nunez, with Cody Gakpo floating between the attack and the right side of the midfield.
With Thiago, Stefan Bajcetic and Jordan Henderson injured or unwell, Klopp may have decided all out attack was the best tactic at his disposal. It almost paid dividends early on, too, but Nunez was denied by Thibaut Courois after Mohamed Salah forced Antonio Rudiger into a slip.
It is fair to say Trent Alexander-Arnold has not had the best time when facing the electric Vinicius Junior over the last few seasons.
The Liverpool right-back has been bested by one of the most devastating wingers in world football on numerous occasions and, yet again, the spotlight was fixed on his performance against the Real Madrid No.20. Alexander-Arnold began the match nervously, taking out Vinicius with a slide tackle at the byline and struggling to locate his intended target while passing.
That set the tone for another occasion when Vinicius got the better of his rival, and he went on to play a key role in Karim Benzema's late goal which killed off the tie.
Although Liverpool had the first clear-cut of the chance through Darwin Nunez, Real Madrid soon imposed themselves on the match in the manner their expectant fans are used to.
Alisson first denied Toni Kroos with a routine save, but there was nothing routine about his spectacular stop to keep Vinicius Junior's effort from close range out following a corner. The Brazilian goalkeeper produced another highlight-reel moment minutes later as he got a palm to Eduardo Camavinga's effort from outside the area to tip it onto the bar.
The next man in pristine white to let fly on the Liverpool goal was Luka Modric, whose fizzing effort sailed off-target as the alarm bells started to ring for the visitors.
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Liverpool ended the first half the team in the ascendency as Real Madrid were seemingly content to let their visitors dominate possession without creating much of note.
Perhaps these encouraging signs convinced Jurgen Klopp not to make any changes at the interval, but it became increasingly evident that Liverpool needed a shot in the arm if they were going to pull off the unthinkable with just over half an hour remaining.
One change the Reds were crying out for saw Roberto Firmino come on for Diogo Jota, but it was the other half of the double substitution - as Harvey Elliott replaced Darwin Nunez - which raised eyebrows. Nunez had been Liverpool's standout attacker throughout the first 57 minutes and his early withdrawal will raise fears that he sustained a knock.
If there was one player who summed up Real Madrid's composed performance, it was Eduardo Camavinga at the base of the hosts' midfield.
The former Rennes prodigy, who is still only 20, was flanked by two Champions League legends in Toni Kroos and Luka Modric but looked at home on the grandest stage of them all. Jurgen Klopp's second-half changes, which also included sending Alex Oxlade-Chamberlain on for James Milner, made no impact on the course of the game.
Los Blancos continued to look unflappable while out of possession, content that Liverpool would fail to play their way through a star-studded defence and a midfield marshalled by one of the brightest talents in European football. Camavinga's cool head was in contrast to Jurgen Klopp's fiery demeanour and the Liverpool manager was booked for protesting against the officials too forcefully.
A deserved breakthrough for Real Madrid eventually came in the 78th minute, when Vinicius Junior's mishit shot fell into Karim Benzema's path and the Frenchman had the simple task to finish into a gaping net.