Ollie Watkins, who scored three goals against Liverpool the last time the sides met, put Villa ahead just before half time. Shortly after, Roberto Firmino had a goal controversially chalked off by VAR. However, it didn't matter in the end as Mohamed Salah and Alexander-Arnold found the net to give Liverpool the win.
"In the end, when you're 1-0 down, the only thing that counts is to score a goal, which we did," Jurgen Klopp said after the game.
"It was a great goal, really nice and the second goal was even nicer. We took a risk with the change -- we brought in an extra midfielder -- and that had a proper impact on both goals. It's exactly what you wish for.
"All we have to do is put the other teams under pressure and win games. We have no time to waste, it's so important to score in the moment and it's exactly why we're doing this."
It took Villa 43 minutes to find a lead through Watkins who hit a left-footed shot past Alisson.
Moments later, Klopp's side were furious when VAR disallowed Firmino's equaliser by the slimmest of margins.
After multiple reviews of the build up to the goal, VAR found that Diogo Jota was offside.
Salah broke a three-game scoreless streak for Liverpool at Anfield on 57 minutes when Emiliano Martinez parried a shot away but only as far as the Egypt international who powered the ball into the net.
His goal is the first from open play that Liverpool have scored in the league at home since Dec. 27.
Liverpool were pummelling the Villa goal in the final moments of the game and it was Alexander-Arnold, one minute into added time, who finally got a shot past Martinez.
The result puts Liverpool in fourth place, equal on points with West Ham who have a game in hand, while Villa are in 10th.