It means Liverpool have not dropped a point in the Premier League since October, when they drew against Manchester United, and keeps them top of the league, 11 points above second-place Leicester City and 17 above Manchester City, who both have a game in hand, Efe news reports.
The Egypt forward's first goal came in the 37th minute following a counter-attack from a weak Watford corner on Saturday.
Roberto Firmino's flick played Sadio Mane in, who slotted it forward to a charging Salah. The Egyptian cut inside Kiko Femenía and curled it into the top of the net with his weaker right foot.
His second goal came just before full time when caught onto a fluffed shot from Divock Origi with a stylish flick to seal Liverpool's victory at Anfield.
Liverpool could have been three goals up if Sadio Mane's header had not been ruled offside by VAR.
Watford's Ismaïla Sarr also missed an opportunity to claw one back with a missed volley within the six-yard box.
On a week when Jurgen Klopp extended his contract with the Reds until 2024, Liverpool remain comfortably ahead in the table with eyes on the next fixture against Astone Villa in the Football League Cup this week.
The Merseyside team will play Leicester in the league on 26 December.
( With inputs from IANS )