Liverpool made an excellent start to the game as Van Dijk headed home from Trent Alexander-Arnold's corner to give the Reds an opener inside the first fifteen minutes.
Roberto Firmino thought he had given the Reds an excellent second with a curling finish into the top right corner, however, the goal was ruled out for a foul on David de Gea by Virgil van Dijk in the build-up.
The Reds saw another goal ruled out after Alex Oxlade-Chamberlain found Gini Wijnaldum in behind the United defence, with the Dutchman's well-placed finish ruled out by the linesman for a marginal offside.
Liverpool dominated Ole Gunnar Solskjaer's side for long periods, especially turning the screw after the break as they piled the pressure on the visitors.
However, a late United resurgence had the Reds on the back foot for much of the closing stages. The Reds survived some nervy moments and finished the game off in injury-time through Salah.
With United camped in Liverpool's half, Alisson took advantage and released Salah through on goal. The Egyptian brushed off Daniel James and slotted past De Gea to seal a 13th consecutive Premier League win for the Reds.
The result sees the relentless Reds go 16 points clear at the top of the table, with a game in hand. Despite some nervy moments, Klopp's side show no signs of slowing down as the league title looks an increasingly likely possibility come May.
Slide through our gallery below for all the best snaps from Anfield.