Desperately in need of victory to keep their top-four hopes alive, Klopp's side deservedly took all three points with a clinical display which has rarely been produced this season.
With fans protesting outside the ground, Bruno Fernandes broke the deadlock after 10 minutes with a deflected strike.
But Liverpool hit back through Diogo Jota who pounced with an instinctive finish.
It only took two minutes into the second-half for Firmino to get his second by leaping on a loose ball after Dean Henderson denied Alexander-Arnold.
Edinson Cavani was uncharacteristically quiet for the first two thirds of the match, but the Uruguayan turned provider to get United back in the game, setting up Marcus Rashford with a deft through pass before the England star finished past Alisson.
However, there was no comeback on the cards this time as Salah put the game to bed with a counterattacking goal on the 90th minute mark.