Salah brought up the milestone in just his 65th game for the club with a second-half penalty for the dominant hosts, having already scored on the stroke of half time to put Liverpool in complete control at the interval.
Jurgen Klopp was able to welcome Roberto Firmino and Sadio Mane back into his starting lineup, and they joined their fellow attacker on the scoresheet with Firmino breaking the deadlock in the first half and Mane capping it off 10 minutes from time.
Mane also missed a penalty, but it did not prove costly as Liverpool cruised to a victory which takes them top of Champions League Group C courtesy of a 2-2 draw between Paris Saint-Germain and Napoli in the other game.
It was Saturday's match-winner Salah who had the first chance of the match when he took Xherdan Shaqiri's lofted ball down before being thwarted by Milan Borjan in the Red Star goal when he only had the keeper to beat.
The visitors held their own for the most part in the opening 20 minutes, though, and could have broken the deadlock themselves when Srdjan Babic took advantage of some statuesque marking to get on the end of a corner, only to fire his volley well over the crossbar.
Liverpool did grab the opener just two minutes later, with Shaqiri threading a pass through for Andrew Robertson, who in turn played a low cross into the box for Firmino.
The Brazilian took his time over his finish and eventually fired a deflected effort past the keeper for his 13th Champions League goal - leaving him just one behind Ian Rush in the club's European history books.
It proved to be a pivotal moment in the match as a previously lacklustre Liverpool suddenly began to play with the same sort of confidence, fluidity and intensity that they showed so often en route to the final of the competition last season.
A second goal almost followed five minutes later when Alisson Becker, Robertson and Mane combined to pick out Salah in the middle, only for the Egyptian to scuff his finish on the volley.
Shaqiri was Liverpool's standout player of the first half and he drew a routine stop from the keeper with a curling effort before Robertson skewed a first-time shot wide when he should have done better having been set up by good work from Salah and Mane.
The second goal did arrive right on the stroke of half time, though, as Shaqiri skilfully cushioned Georginio Wijnaldum's pass into the path of Salah, who lashed his right-footed finish past the keeper at the near post for his first of the night.
The half-time scoreline reflected Liverpool's dominance since breaking the deadlock, and they were in no mood to let up in the second half, with Fabinho firing off target on his first Champions League start for the club to give an early reminder of the hosts' intentions.
It took just five minutes of the second half for the Reds to effectively kill the game off, although their third goal was not short of controversy as Mane went down dramatically under the challenge of Filip Stojkovic to draw a belatedly-awarded penalty from the officials.
It was a soft one for the hosts, but it led to a historic moment as Salah drilled the spot kick straight down the middle to bring up his half-century from just 65 appearances for the club - smashing the previous record of Fernando Torres, who took 72 games to reach that tally.
Liverpool, who beat Maribor 7-0 on matchday three of last season, began looking as though a similar scoreline could be on the cards this time around too and Trent Alexander-Arnold almost got in on the act when he latched on to Fabinho's exquisite pass over the defence but could not lift his finish over the keeper.
Salah squandered one half-chance to complete his hat-trick when he failed to connect with an attempted snapshot on the turn, and Firmino stung the palms of Borjan with a drilled effort from range moments later as the hosts maintained their relentless pressure.
Salah's chances of a treble were ended when he was replaced by Daniel Sturridge with just under 20 minutes remaining, but Liverpool's attacking threat did not leave with the Egyptian record-breaker and just three minutes later they were awarded a penalty for a handball against El Fardou Ben Nabouhane.
Klopp may have wished that he had held on for just a while longer before taking Salah off, though, as Mane stepped up to take the spot kick and saw his effort tipped onto the crossbar before poking his rebound wide of the target.
The Senegal international made amends just four minutes later when he did join his fellow forwards on the scoresheet, poking his low finish past the keeper after being teed up by Sturridge to end a seven-game goal drought.
It should have been five just a minute later when Sturridge sent Adam Lallana clean through on goal, but Borjan stayed big to deny the sub his first goal of the season.
Liverpool kept on pushing until the final whistle, though, with Sturridge testing the Red Star keeper himself in the 90th minute before Firmino played Mane in on goal and the winger fired his finish into the side-netting.
In the end, the four-goal winning margin flattered Red Star more than it did Liverpool, who bounce back from their defeat to Napoli on matchday two to take the lead in Group C, now one point clear of the Italian outfit and two above PSG.
Group-stage debutants Red Star remain bottom on one point, meanwhile, having now conceded 10 goals in their two Champions League away games following their 6-1 defeat to PSG on matchday two.
LIVERPOOL (4-2-3-1): Alisson; Alexander-Arnold, Gomez, Van Dijk, Robertson (Moreno 82'); Wijnaldum, Fabinho; Shaqiri (Lallana 68'), Firmino, Mane; Salah (Sturridge 73')
RED STAR (4-2-3-1): Borjan; Stojkovic, Degenek, Babic, Gobeljic; Krsticic, Jovicic (Causic 75'); Ben Nabouhane (Simic 81'), Ebecilio (Jovancic 65'), Srnic; Boakye body check tags ::