Juergen Klopp's side, who have won 10 of their last 12 matches against Villa, climbed to third in the table on 10 points from four games. Villa have six points.
Salah, reportedly a transfer for Saudi Arabian club Al-Ittihad, netted goal number 188 of his Liverpool career with a toe poke at the back post in the 55th minute.
Szoboszlai had given Liverpool a dream start with a left-footed half-volley from the edge of the 18-yard box after three minutes. Liverpool doubled their lead when Darwin Nunez's firm shot hit the post and rebounded into the net off Villa defender Matty Cash.
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