Jurgen Klopp's side created enough to chances to turn the tie on its head but their experienced opponents kept their heads brilliantly to see out the tie.
Real will now face Chelsea in the last four as they look to secure yet another European Cup under Zinedine Zidane.
See how the players rated from an interesting tactical battle below.
✔️ Casemiro: 7.6
While Luka Modric and Toni Kroos struggled to exert the same level of control they enjoyed in the first leg, Casemiro seemed to relish being involved in a completely different type of game.
The 29-year-old showed he was up for the battle with a retaliatory challenge on James Milner early in the second half but was more measured and intelligent with his tackles from there.
His contributions helped Real keep some semblance of grip on the game.
❌ Sadio Mane: 6.0
Something of a surprise pick given his poor recent form, Sadio Mane looked back to his best during an opening 10 minutes that saw makeshift right-back Fede Valverde struggle.
But the Senegalese could not sustain his momentum and fell away badly as Liverpool failed to capitalise on their chances.
⭐ Nacho: 7.7
The Spaniard was also noticeably the most vocal member of a backline that had to stay organised in the face of a stern examination, particularly in the second half.