Everton are currently seventh in the Premier League with 24 points while Watford are 12th with 21 points.
City's goals came either side of half-time as Leroy Sane chested home before Riyad Mahrez doubled the visitors' lead early in the second period.
Unai Emery's side were under the cosh in the second half at the Emirates Stadium.
United have endured a miserable start to the season, losing two of their first four matches and slipping well off the searing pace set by Liverpool, Chelsea and Watford.
Liverpool will face their first serious test of the Premier League campaign when they visit Tottenham Hotspur on Saturday.
Chelsea opened their Premier League campaign with victory against Huddersfield, while Tottenham, Bournemouth, Crystal Palace and Watford also made winning starts.
Tottenham Hotspur edged closer to a third consecutive season in the Champions League by opening up a five-point lead on fifth-placed Chelsea.
Klopp believes Salah, 25, has time on his side to succeed Messi and Real Madrid rival Cristiano Ronaldo, both now in their thirties, as the world's best player.
Liverpool moved a step closer to securing Champions League football next season via a top-four finish.
Arsene Wenger's side ended a run of three successive Premier League defeats with a solid display at the Emirates Stadium.
Goals from Daryl Janmaat, Gerard Deulofeu and Roberto Pereyra in the final six minutes handed Watford a welcome first win in six games to move the Hornets six points clear of the relegation zone.
Watford said Everton's approach for Marco Silva had been "the catalyst" for the decision.
Raheem Sterling scored the fastest goal of the Premier League season as leaders Manchester City got back on track with a 3-1 win against Watford on Tuesday
Moyes was in charge of West Ham for the first time following his appointment as replacement for the sacked Slaven Bilic.
Watford striker Troy Deeney will serve a three match ban for violent conduct during last Saturday's fiery Premier League clash with Stoke, the Football Association announced on Tuesday.
Aaron Mooy and Laurent Depoitre scored at the John Smith's Stadium to give Huddersfield a first win over United since 1952 and condemn Jose Mourinho's side to their first loss of the campaign.
Sergio Aguero's hat-trick inspired a 6-0 demolition of Watford.
Nathaniel Chalobah can operate in midfield or defence and played 10 times for Chelsea in their Premier League title-winning campaign last season.