Early first-half goals from Douglas Luiz and Joel Matip left the match level at the break, with Sadio Mane netting a crucial winner in the 65th minute to hand Liverpool the three points and keep their hopes of beating City, who have a game in hand and a superior goal differential, to the title.
Liverpool, who have two matches left, went behind before five minutes when Luiz pounced on the rebound from his own saved header and blasted the ball into the roof of the net to hand Villa an early lead.
Matip put Jurgen Klopp's side back on level terms minutes later, pouncing on a shot from Virgil van Dijk that was blocked into his path and reaching out to poke home to make it 1-1.
The visitors, uncharacteristically nervy, suffered a setback around the half hour mark as defensive midfielder Fabinho was forced off injured as the first half finished level.
Mane -- who has a torrid record against Aston Villa with nine goals in 10 games against them -- popped up in the second half to head home an assist from Luis Diaz to give Liverpool a lead they would hold on to. It was his 15th league goal of the season, the third Liverpool player to reach at least that mark this campaign after Mohamed Salah and Diogo Jota.
"He's a machine, I told him after the game. He is a massive player. His physicality is brutal. He is a mix of technique, desire and physicality when at the moment everyone has a few yards in the legs," Klopp said. "You can see it with him to be honest. Top goal, he is just a fantastic world-class player."
Liverpool, who can still win the Quadruple, have a busy end to an already packed campaign, playing Chelsea in the FA Cup final on May 14 and Real Madrid in the Champions League final on May 28 along with their remaining Premier League matches against Southampton and Wolves on May 17 and 22, respectively, in between.
"It was always going to be tough and we have to keep going. We're in the same position as we were before the game. It was a big game and another three points," Liverpool captain Jordan Henderson told Sky Sports.
"We concentrate on what we need to do. It is another big game at the weekend, the FA Cup final and we have to recover for that. It is out of our control [the title race], we do not need to waste energy on that."
Steven Gerrard's Villa remain in 11th place on 43 points with three games left to play starting with Crystal Palace on Sunday.