Liverpool was energized by Manchester City's recent defeat, and Nunez's goal in the 20th minute set the tone for the match at Anfield with a lightning-fast counter-attack.
Liverpool goalkeeper Caoimhin Kelleher made some vital saves to preserve his side’s lead but Salah raced clear in the 84th minute to wrap up the points and maintain their sensational start to the season under new manager Arne Slot.
They have won 15 out of 17 games in all competitions since the Dutchman replaced Juergen Klopp in the close season.
Slot’s side have 28 points from 11 games with reigning champions City on 23.
Villa’s third league defeat of the season leaves them down in eighth place with 18 points.