Alexis Mac Allister headed in the only goal but Madrid were grateful to goalkeeper Thibaut Courtois for keeping the score down.
Trent Alexander-Arnold was met with a hostile reception on his return to Liverpool as a late substitute and the home crowd roared the Reds over the line for what could be a turning point in their season.
Both sides are now tied on nine points from their opening four games of the league phase.
Liverpool had lost six of their previous eight games in all competitions, but stopped the rot by beating Aston Villa on Saturday as Arne Slot resorted to a more tried and trusted team selection.
Florian Wirtz replaced Cody Gakpo in the only change from the weekend and Liverpool again looked more like the side that stormed to the Premier League title last season.
“It is easy to say now because we won twice in a row,” said Liverpool captain Virgil van Dijk. “In a world of chaos you have to stay calm and see perspective of things. We all know how football works. It can change overnight.”
Madrid had lost just twice in 20 games since former Liverpool favourite Xabi Alonso took charge but relied on Courtois to avoid a similar scoreline to their only previous defeat this season – 5-2 at Atletico Madrid.
“Nothing to reproach the team for. It is only the league phase and we can recover these points,” said Alonso. “We lost due to small details. We lacked threat in the final third.” (AFP)
