Jurgen Klopp's side collected their first home league win in 2021 after right back Alexander-Arnold scored with a curling effort from the edge of the box at the death.
The win moved Liverpool up to fourth, on 52 points, above West ham United on goal difference, yet their stay in the top four will be short-lived if Chelsea beat Crystal Palace in the final game of the day.
Having suffered a humiliating 7-2 defeat at Villa Park in October, defending champions Liverpool enjoyed a better start at Anfield as Mohamed Salah and Diogo Jota tested visiting goalkeeper Emiliano Martinez with early attempts.
But Villa, against the run of play, found the breakthrough in the 43rd minute when John McGinn slotted a neat pass into striker Ollie Watkins, whose drilled effort slipped under Liverpool keeper Alisson's right arm.
Liverpool responded almost immediately through Roberto Firmino but his goal was chalked off by VAR as Jota was adjudged to be marginally offside in the buildup.
Yet the hosts deservedly equalised 12 minutes into the second half through a Salah header after left back Andy Robertson's initial strike was palmed away by Martinez.
It was Liverpool's first goal in open play at Anfield since December 27 - a run of run of 12 hours and 44 minutes since Sadio Mane found the net against West Bromwich Albion. They netted one home goal from the penalty spot during that time.
Klopp's side stepped up the intensity and Alexander-Arnold's sublime effort secured all three points.
"We have lost a couple of games here and mentally it was like 'oh no, not again' but we did a good job second half," Salah told Sky Sports.
"It's really important to win. I think there are only seven games left? We want to play Champions League next year but we just need to focus on each game and try and win them all and hopefully, we can be in that fourth place."
Villa boss Dean Smith, who was once again without the injured playmaker Jack Grealish, conceded his players struggled to create enough goalscoring opportunities in the second half as they remain 10th in the table.
"We had some good opportunities but didn't have that final pass," Smith said.
"We've come to Liverpool, knew how they would play and how determined they were to put things right at Anfield but it's disappointing to concede so late." (Reuters)