The Reds trailed by a three-goal margin ahead of the second leg of the last-16 contest and were unable to overturn the deficit, losing 6-2 on aggregate.
Karim Benzema struck the decisive goal on the night in an even contest, converting from close range 12 minutes from time to settle the encounter in the hosts’ favour.
Team news
Diogo Jota and James Milner moved back into the Reds’ starting line-up for the match, replacing Stefan Bajcetic and Harvey Elliott.
Liverpool: Alisson, Alexander-Arnold, Konate, Van Dijk, Robertson (Tsimikas, 90), Fabinho, Milner (Oxlade-Chamberlain, 73), Gakpo (Carvalho, 90), Nunez (Firmino, 57), Salah, Jota (Elliott, 57).
Unused subs: Adrian, Kelleher, Gomez, Keita, Jones, Matip, Williams.
First half
The hosts looked to threaten early on though Benzema was flagged offside as he burst through on goal to see a shot saved by Alisson Becker.
In similar fashion, Mohamed Salah played in Darwin Nunez at the opposite end after stealing the ball in the home half, but the forward had a low effort blocked by the legs of the onrushing Thibaut Courtois.
Alisson again intervened as he saved from Toni Kroos and Vinicius Junior, before tipping an Eduardo Camavinga effort onto the bar.
Cody Gakpo glanced a header wide of the mark in response, while Nunez forced a fine save from Courtois’ fingertips as he denied the dipping effort.
Ahead of the half-time interval, Gakpo again tested the Real stopper, with the home No.1 reacting to a well-struck half-volley.
Second half
Liverpool began after the interval on the front foot but it was Real who forced the first save of the half.
Alisson once more thwarted the advances of the hosts, blocking from Federico Valverde and then Benzema.
The former also guided a header over the bar a short while later following good work by Luka Modric down the left, while the latter lifted a shot from inside the box over the frame of the goal.
It was the hosts’ No.9 who eventually made the difference in the 78th minute, with Vinicius prodding a loose ball into his path inside the box to allow a close-range effort to be converted.