Firmino was perfectly placed to open the scoring on 31 minutes when he met Alexander-Arnold’s pinpoint cross from the left to send a header past goalkeeper Kasper Schmeichel.
Liverpool moved through the gears with three second-half goals to put the game out of reach.
Substitute James Milner made it 2-0 after 71 minutes, scoring with his first touch from the penalty spot after Caglar Soyuncu had handled Alexander-Arnold's corner.
Three minutes later another inviting delivery from Alexander-Arnold was swept home by Firmino to make it 3-0.
The impressive Alexander-Arnold added his side's fourth goal with a powerful low finish set up by Sadio Mane.
Leicester: 28 Dec v West Ham (A)
Liverpool: 29 Dec v Wolves (H)