The Egyptian King set up Sadio Mane for a ninth-minute opener with a wonderful diagonal pass with the outside of his foot before scoring himself on 54 minutes to make it 4-0.
He skipped past three challengers in the area, flummoxed one of them with a superb piece of individual skill to leave him on the floor before planting a delightful curler into the far corner.
Salah took his tally to 104 Premier League goals, on level with Ivorian great Drogba, who reached that mark over two spells with Chelsea.
Roberto Firmino also hit a hat-trick as Liverpool ruined Claudio Ranieri's first match as Watford manager.
Salah underlined why Liverpool are so desperate to get the Egypt forward to sign a new contract with his audacious second-half strike at Vicarage Road.
Dancing through the Watford defence, Salah's magical moment topped even his eye-catching goal against Manchester City before the international break.
Liverpool's habit of avoiding big contracts for players over 30 has fuelled fears the 29-year-old will leave when his present deal expires in 2023.
But with 10 goals in 10 games this season, Salah has proved his huge value to Liverpool, who will surely bow to his wish to be paid alongside the Premier League's highest earners.
For the first time in his career, Salah has scored in eight successive top-flight matches.
Salah's breathtaking goal stole the spotlight on a day when Firmino claimed his first Liverpool treble since 2018 in his first start for five league games.
It was also a memorable afternoon for Senegal forward Sadio Mane, who became only the third African player after Salah and Didier Drogba to reach a century of Premier League goals, a milestone encompassing his time at Southampton.
The Reds are the first English top-flight side to score three or more goals in seven consecutive away games in all competitions.
Liverpool, unbeaten in eight league matches this season, moved above Chelsea to the top of the table, although the Blues can reclaim pole position if they beat Brentford later on Saturday.
Ranieri could not have asked for a tougher start to his reign after he replaced the sacked Xisco Munoz during the international break.
The Italian, back in the Premier League after his title-winning spell with Leicester and stints at Chelsea and Fulham, will hope this dismal display was not a sign of things to come.
Having drawn their previous two league matches before the international break, Liverpool couldn't afford another slip and they tore into Watford from the start.
Liverpool held over 80 percent of the possession in the first half, with Salah tormenting Danny Rose time and again down the right flank.
It was no surprise when Klopp's team took the lead in the ninth minute.
Salah nudged Rose aside with ease and unfurled a glorious pass with the outside of his boot that evaded the entire Watford defence as it arrowed towards Mane, who applied the finishing touch with a first-time shot into the bottom corner.
It was almost too easy for Liverpool and Firmino was the next beneficiary of the visitors' supremacy in the 37th minute.
A superb move cut through the Watford defence as Mane's deft pass found James Milner's overlapping run and his cross was perfectly weighted for Firmino to tap in his fourth goal this season.
A deflected drive from Liverpool's Naby Keita hit the bar just before a half-time whistle that was greeted with boos from frustrated Watford fans.
Unable to muster a single shot in the first half, Watford's backs remained against the wall and Salah's curler was tipped away by Ben Foster.
Liverpool were showing no mercy and Firmino notched their third goal in the 52nd minute.
Salah was in an offside position when Andrew Robertson crossed, but Craig Cathcart's sliding intervention deflected the ball to Firmino, who finished from close range.
If that was a no-frills strike, Salah's 54th-minute goal will be remembered for years to come.
Collecting Firmino's pass in the Watford area, Salah swivelled away from three defenders with an audacious ball-roll, then left Cathcart on his backside before guiding a superb finish into the far corner.
Firmino completed the demolition with a simple finish from Neco Williams' cross in the final seconds.