Mohamed Salah has scored two goals in four games so far this season
Jurgen Klopp's side rejected a £150m bid for the Egypt international from Saudi Pro League club Al-Ittihad, with the Reds insisting the 31-year-old is not for sale.
And Salah emphasised his worth to Liverpool when he scored their third goal in the second half at Anfield, firing in at the back post from Darwin Nunez's cross to seal a third league win in a row.
Dominik Szoboszlai got the hosts up and running after just three minutes, scoring his first goal for the club with a crisp strike through a crowded box.
Villa, who have been scoring for fun since losing 5-1 to Newcastle in their opening game of the season, were far too open at the back as they struggled to deal with Liverpool's forwards.
The hosts increased their lead in the 22nd minute when Nunez's shot hit the post and bounced in off the unlucky Matty Cash.
Liverpool looked capable of scoring with every attack in a rampant first half, although Villa's cause was not helped by an enforced change as centre-back Diego Carlos limped off with an injury inside the opening 20 minutes.
And the lively Nunez went close to scoring his third goal in two games as his shot bounced off the crossbar.
Keeper Alisson made a good stop to push out Cash's header in the second half, but Salah's second goal of the season wrapped up a win that moved Liverpool up to third in the Premier League, two points behind leaders Manchester City.
-- Courtesy of BBC Sport