The Reds, comfortably defeated their rivals - who had won their previous two consecutive games, thanks to Alexander-Arnold's double and further second-half goals from Paulo Alves and Corey Whelan, after goalkeeper Ollie Byrne saw red for the hosts at the Aon Training Complex.
After a cagey opening 20 minutes, in which opportunities were few and far between Zachary Dearnley came closest to breaking the deadlock as his effort, after cutting in from the left, crash off the post and roll across the goal-line.
The visitors had gone close after Adam Phillips' audacious lob from the halfway line flew over the crossbar, but had otherwise struggled to create any real opportunities.
But they took the lead after 26 minutes thanks to skipper Alexander-Arnold, who combined with Ben Woodburn before headed past Byrne to open the scores after Tosin Kehinde had been dispossessed.
Paul McGuinness' home side struggled to get back at their rivals, who held their own until half-time to preserve a single goal advantage.
Sublime second-half sees Reds run amok
The Reds doubled their lead around the hour mark after some brilliant play by Alves saw him put them well in command at Carrington.
The Portuguese attacker spun his marker at the edge of the box before hitting a low strike into the bottom corner with power to make it 2-0 to Liverpool.
Then, in a game-changing moment - Marcus Rashford was denied by the woodwork after he had broken free of his opponent and struck a shot from the edge of the box.
Almost immediately up the other end, goalkeeper Byrne was sent off for a second yellow card after rushing off his line and conceding a foul for the second time in the game, bringing down Yan Dhanda.
From the following free-kick, Liverpool made sure of the points as Corey Whelan rose unopposed to head past substitute shot-stopper George Dorrington.
Neil Critchley's side weren't done there, adding another in the final stages to put the gloss on the game - as Alexander-Arnold grabbed his second of the game with a side-footed finish from Woodburn's cross, taking their fourth win from seven to move into 5th in the table.