The Egyptian put Liverpool ahead after only four minutes, dribbling past Miguel Britos to slot home.
His second came from Andrew Robertson's cross, which was converted at the far post.
Watford's hopes of a comeback after the break were hit when Roberto Firmino flicked in Salah's cross on 49 minutes.
Salah's first Premier League hat-trick came when he twisted this way and that before poking home on 77 minutes.
He then followed up Danny Ings' saved shot to move to 28 goals for the season, four ahead of Harry Kane.
The win sent Liverpool two points above Tottenham Hotspur into third, two behind Manchester United.
Watford drop to 11th, on 36 points.
Liverpool: 31 Mar v Crystal Palace (A)
Watford: 31 Mar v AFC Bournemouth (H)