After a passive first-half, it was Salah who took hold of the game in the second stanza when he collected a pass from substitute Curtis Jones before cutting in on his left foot and dispatching a curled strike into the top-corner.
Liverpool's second was a piece of art and will be shown time and time again when the 2020/21 season's best goals are aired in years to come.
Breaking on the counter-attack, Trent Alexander-Arnold pinged a 40-yard diagonal ball to unleash Xherdan Shaqiri on the left-wing. The Swiss maestro then, instead of travelling into acres of space, played a brilliant looping first-time pass to Salah, who took a touch to control before finishing smartly past Lukasz Fabianski.
To prove the second goal wasn't a fluke, Liverpool's third was also an ingeniously swift team goal, reminiscent of Klopp's side from last season.
They will have a chance to enter the clash with three straight wins if they can beat Brighton at Anfield on Wednesday.
The Hammers now have their first loss for 2021 but still remain in the hunt for a spot in Europe next season.
Speaking after the game, Klopp said of his team's performance: "We didn't create enough in the first half. We made little adjustments at half-time and it paid off. We scored three unbelievable goals.
"These boys are a really good group. They're not happy at not being successful, not winning, but they always have the right attitude.
"We slept one night at home [after Tottenham] and then we came to London. That's two great results."