Liverpool moves into third place, with a point behind Manchester United and four behind Manchester City who has a game in hand.
The Reds will face Brighton next as they look to make it three wins in a row.
West Ham remains fifth but is likely to be leapfrogged by Tottenham who sits two points behind with two games in hand.
Moyes will be hoping his side can bounce back to winning ways when they take on Aston Villa next week.
Three brilliant second-half goals from the Premier League champions means they leave the London Stadium with a 3-1 win over West Ham.
After a fairly uneventful opening 45 minutes, Salah broke the deadlock in the 57th minute when he curled an effort into the far corner.
And the Egyptian doubled Liverpool’s lead not long after with another incredible goal at the end of a lightning-fast counter-attack.
Wijnaldum made it three in the 84th minute before Dawson’s consolation goal took some of the glow of the performance.
Liverpool has played here five times since West Ham moved into their new stadium: won four, drawn one. Salah has played in four of those games took his goal tally to five at London stadium.
It is fair to say Liverpool’s blip is over, with back to back victories away to the capital’s highest-placed teams.
West Ham was on an unbeaten run of eight games, and six wins but was significantly second best here. Salah’s two second-half goals gave the weary team a vital win to start the week which concludes with Manchester City’s visit to Anfield.
Salah was the only fellow of Liverpool’s usual forward three to start, but he was enough.
Liverpool’s second came from a West Ham corner, in 68 minutes the wild counter-attacking of old, all speed, and pinpoint passing.
A clearance to Alexander-Arnold, a precise cross-field ball to Shaqiri, a stunning cross to the feet of Salah, with one touch to control, as he dinks the ball into the net. Beautiful.
West Ham was done by then, after 84 minutes, Alexander-Arnold involved again before Roberto Firmino laid the ball for Georginio Wijnaldum to finish.
Craig Dawson ran one in from a poorly-defended corner soon after but it gave the scoreline a dented look it did not deserve.
West Ham’s only decent chance came after 24 minutes when an Aaron Cresswell cross should have been finished by Said Benrahma, but he hesitated as he pass the ball to Pablo Fornals, whose shot was headed off the line by Andrew Robertson.
Liverpool was in a different class here. It is hard to see West Ham as top-four contenders on this evidence.