Juergen Klopp’s side, chasing their first league title in 30 years, also have a game in hand on second-placed Manchester City with the title race appearing more of a procession than a contest.
For United, who battled hard but were outclassed in all departments, it was another reminder of just how much work is ahead of them if they are to get back to being in contention for titles.
United boss Ole Gunnar Solskjaer suffered another blow to his hopes of qualifying for the Champions League with the news that his top scorer Marcus Rashford was not only out for the trip to Anfield but could face several weeks on the sidelines with a back injury.
Some reports even suggested he might be out for up to three months.
United are in fifth place in the league, five points behind fourth-placed Chelsea, and while no-one could question the commitment of Solskjaer’s players it is clear he needs some immediate transfer reinforcements.
After a promising start from the visitors, Van Dijk put Liverpool ahead with a firm header after he was poorly marked by United, who allowed him to rise and nod in a Trent Alexander-Arnold corner.
Roberto Firmino thought he had scored a second but a VAR review ruled that Van Dijk had fouled United keeper David de Gea before the ball fell to the Brazilian.
Liverpool had the ball in the net again when Georginio Wijnaldum slotted home after a clever defence-splitting pass from Alex Oxlade-Chamberlain but the Dutchman was ruled narrowly offside.
United’s attacking forages were rare but they caused some anxiety for Klopp when Andreas Pereira slid in but could not make contact with a low ball across the face of the goal from Aaron Wan Bissaka.
Yet after the break, Liverpool looked determined to put the outcome beyond doubt.
Salah missed a glorious chance from close range just after the interval during an intense period of pressure from the home side, in which Jordan Henderson also hit the post.
United survived that blitz and clawed their way back into the game and could even have snatched an equaliser but Anthony Martial blasted high and wide in the 59th minute.
Solskjaer’s side applied some late pressure but were caught on the counter-attack in the final moments when, with all United players forward, Liverpool keeper Alisson Becker launched a long ball for Salah who raced clear before slotting home and settling Anfield’s nerves.
“It’s special. We’re enjoying our football. You have to stay hungry and keep wanting more,” said Henderson, who looks well set to be raising another piece of silverware at the end of this campaign after lifting the Champions League trophy last season.