Jurgen Klopp's side struggled to find a way through the Hornets defence for 66 minutes on Saturday, the Reds perhaps having an eye on Wednesday's pivotal Champions League game away to Paris Saint-Germain.
The hosts could have taken all three points, Gerard Deulofeu having an early goal ruled out for offside while a strong penalty appeal from Will Hughes was denied by the referee.
But Salah hit a first-time finish through the legs of goalkeeper Ben Foster, scoring his fourth goal in five Premier League games, before Alexander-Arnold's wonderful free-kick put the visitors in control.
Their win was marred, though, when captain Jordan Henderson was dismissed with eight minutes to go, meaning will be banned for next weekend's Merseyside derby, although Firmino added a late third.
Simply. Unstoppable. pic.twitter.com/Z3Jej0nWcz— Liverpool FC (@LFC) November 24, 2018
Watford had a goal ruled out in the second minute, Deulofeu running on to Troy Deeney's pass to beat Alisson with a smart finish, but he was offside.
Liverpool added Xherdan Shaqiri to a front three of Salah, Firmino and Sadio Mane – who signed a new deal on Thursday – but it took them 38 minutes to record a shot.
Firmino found space and fired off a low effort which Foster held well, Roberto Pereyra then testing Alisson's reflexes at the other end before the Watford goalkeeper kept out Salah's header.
Referee Jon Moss waved away Watford penalty appeals when Hughes went down under Andy Robertson's rash challenge early in the second half.
Salah should have opened the scoring, slotting his finish through Foster's legs but wide of the far post, but he did break the deadlock soon after.
Firmino fed Mane through the left channel and the Senegal star pulled back a low cross for Salah, whose finish lacked power but was directed well enough to beat Foster.
Alexander-Arnold put Liverpool's firmly in the ascendancy with 14 minutes to go, curling a fantastic 25-yard set piece out of Foster's reach for his first Premier League goal of the year, before Henderson was sent off for a second yellow card late on.
And Firmino added his name to the scoresheet in the 89th minute, nodding in from close range after Foster denied Mane.
39 - Mohamed Salah is Liverpool's highest Premier League goalscorer under Jurgen Klopp, despite playing in just 41% of their games since the German arrived (49/119). Remarkable. #WATLIV— OptaJoe (@OptaJoe) November 24, 2018
What does it mean? Reds still unbeaten
Avoiding defeat for the first 13 games of the season is the longest unbeaten run for Liverpool at the start of a Premier League campaign since 2007-08. For much of this game the Reds lacked their usual attacking fluidity, but in Salah they have a goalscorer who came up trumps again.
Firmino provides creative spark
Mane's new contract means all three of Liverpool's strike force have committed their futures to Anfield this year, but it was Firmino who managed to unlock a determined Watford defence, his pass to feed Mane ultimately teeing up Salah's goal.
Henderson lets his team down
Liverpool captain Henderson was starting a Premier League game for the first time in a month, but he appeared off the pace at Vicarage Road. A mediocre day for the England midfielder was made worse when he saw red late on for a barge on Etienne Capoue having already been given a final warning by Moss.
Key Opta Facts
- Watford have lost three of their last four home top-flight games (W1), as many defeats as they suffered in their previous 15 at Vicarage Road in the Premier League (W8 D4).
- Liverpool are unbeaten in their opening 13 matches of a Premier League season for the first time since 2007-08.
- Watford have lost 16 of their 18 Premier League matches against sides starting that day in the top two of the table, beating Chelsea in September 1999 and Arsenal in January 2017.
- Liverpool's Mohamed Salah has netted six goals in his three Premier League appearances against Watford.
- Mohamed Salah has had a hand in six goals in his last five Premier League games for Liverpool (four goals, two assists), as many as he managed in his previous 11.
Liverpool have a crucial Champions League clash with PSG on Wednesday before hosting Everton in the Merseyside derby, when they will be without captain Henderson due to suspension. Watford are at Leicester City next Saturday.