The Egyptian scored twice after Roberto Firmino had opened the scoring in the 20th minute to set Jürgen Klopp's men on their way with a deflected effort.
Salah got his first of the night on the stroke of half time, doubling up from the penalty spot six minutes into the second half. Sadio Mané rounded off the scoring late on after missing a dubious penalty minutes before.
Here's what we learned from a routine victory at Anfield…
This is Klopp's Liverpool 2.0
Few coaches are as synonymous with a particular tactical approach as Klopp. In theory anyway. In reality the Reds' charismatic coach is far more flexible than some would have you believe yet he is famed for gegenpressing and ‘heavy metal football'.
But seasoned Liverpool watchers will tell you the Reds have actually adapted how and when the press since Klopp has been at Anfield.
With Liverpool flattering to deceive so far this season – at least against the extraordinarily high standards they set last season – there have already been questions. Have they been found out? Did Salah simply enjoy a career season he can't possibly hope to repeat? Why aren't Liverpool as fun as last season?
Klopp will just be happy to finally be winning the dull games. The Reds found a way against Brighton & Hove Albion on Matchday three and were 1-0 winners at relegation fodder Huddersfield Town on Saturday. It's not all about 4-3 wins over Manchester City, after all, and results against lesser sides have previously been their undoing.
Evidently Klopp feels he can get more from this side, though. Having first toyed with a 4-2-3-1 against Southampton Klopp returned to that page of his playbook for the visit of Red Star Belgrade. Sweeping conclusions cannot be drawn after 90 minutes against one of the Champions League's weaker teams but the system, nominally at least, offers Liverpool a different option…
…Firmino moves into Number 10
The Brazilian has played in virtually every attacking position during his Liverpool career – and some defensive ones too, most notably at right wing-back against Arsenal under Brendan Rodgers.
This season, though, he has taken on a new role as a false nine, dropping deeper to allow Salah to play through the middle on the shoulder of the last defender where his direct running and pace make him most dangerous.
It would have been easy for Klopp to keep calm and carry on, sticking to the 4-3-3 which carried Liverpool all the way to the Champions League final last season. But the former Dortmund coach is more proactive than that and has tinkered with his side in order to get the best out of his star man, Salah.
That has meant Firmino dropping back into midfield to provide the link between midfield and attack. Selfless and tireless in his work-rate, the Brazilian is the ideal man for the role and, as shown in his pass map above – courtesy of Stats Zone – did the bulk of his work well outside the penalty area.
Good things come to those who wait
Fabinho has had to be patient since joining Liverpool for £39million from Monaco in the summer. So patient, in fact, that given he has just 21 minutes of Premier League action to his name he would be forgiven for wondering if he'd made a mistake in moving to Anfield in the first place .
But Liverpool and Klopp have been here before. It took Alex Oxlade-Chamberlain some time to work his way into the Reds' starting XI last season. Ditto Andrew Robertson. Both were soon established members of the side before long and both appeared to benefit from being given a grace period to adapt to the demands of playing for Klopp.
Moving to a new club is one thing. But learning to play in a Klopp team is even more daunting. The German demands total assimilation from his players and that takes time – even with pre-season to bed in. Fortunately Klopp has built up enough goodwill he doesn't need to rush players in and Fabinho is the latest beneficiary.
After a slightly slow start the Brazilian grew in confidence and influence alongside Gini Wijnaldum in Liverpool's new-look 4-2-3-1. He got forward at opportune times, making space and creating danger; using the ball neatly and showing a defensive steel when Red Star threatened to break.
In the first half alone he won four of five tackles, recovering possession three times and completing 34 of 38 passes.
Shaqiri continues to grow into Reds shirt
Another summer signing who has had to make do with fleeting action is Xherdan Shaqiri. Finally handed his first Premier League start for the Reds against Southampton in September he was promptly hooked at half time despite causing the Saints untold problems and having a hand in all three goals.
Klopp was believed to be unhappy about the Swiss maverick's lack of positional discipline and failure to help his team-mates out defensively. But that is not why you sign Shaqiri. The diminutive attacker's qualities lie in his unpredictability and eye for the unexpected.
Handed the right-wing role in Klopp's new-look 4-2-3-1 the former Internazionale and Bayern Munich playmaker shone. He supplied the pre-assist for Firmino's opener and also the pass for Salah to double up before half time.
By the break Shaqiri had created two big chances and completed eight of the ten passes he'd attempted in the final third. Whether or not Klopp can turn Shaqiri into the well-rounded player he so desires or not remains to be seen. But as a creative force, the pocket playmaker is an asset.
Robertson: The Flying Scotsman
If Shaqiri needs any reassurance about the merits of playing the long game then he need look no further than the example set by Robertson.
The new Scotland captain started last season behind Alberto Moreno in the pecking order following his move from Hull City. The same Moreno who had been eviscerated by Gary Neville for his part in Liverpool's UEFA Europa League final defeat to Sevilla in 2016.
At that point it looked a long way back for Moreno. So when Robertson started just twice in the Premier League before December it appeared as though he might have come up short against Klopp's expectations for him.
But since dislodging Moreno in December 2017 he has not looked back, establishing himself as one of the Premier League's finest left-backs. Unlike plenty of modern full-backs he can defend. But going forward he is a constant threat, too, and against Red Star his full attacking arsenal was evident.
Gunning down the left flank at every opportunity, Robertson's low cut-back found Firmino to open the scoring and in total he completed 16 passes in the attacking third (fourth overall).