Defeat by Napoli last time out made this a must-win game in the battle with the Italians and Paris St Germain for the qualifying spots, and Liverpool’s class told as they ran out 4-0 winners.
Roberto Firmino continued his fine scoring record in the competition with the opener in the 20th minute before Salah took over, netting his first just before half-time and then scoring a penalty shortly after the break to make it 50 goals in just 65 Reds appearances.
A second penalty was missed by Sadio Mane but the forward made amends moments later to net Liverpool’s fourth.
Klopp stressed in his programme notes that Red Star should not be underestimated but the Serbian champions, playing in the competition proper for the first time since it became the Champions League in 1992, looked clearly the weak link in Group C.
Liverpool welcomed back Mane following his broken thumb but their injury problems persist and Virgil van Dijk captained the team in the absence of the hamstrung Jordan Henderson.
Brazilian midfielder Fabinho made only his second start for the club and impressed in moments while also giving the ball away too cheaply.
The early stages showcased the “average” Liverpool that Klopp referred to after they ground out victory over Huddersfield at the weekend, with passes misplaced and Red Star looking the more threatening.
The home side’s opening 10 minutes was summed up by Trent Alexander-Arnold inadvertently blocking Firmino’s shot when Liverpool finally created something of note.
Moments later Xherdan Shaqiri played Salah clean through but, although his first touch was good, the Egyptian dallied too long before shooting and keeper Milan Borjan blocked.
After 19 minutes of little quality, a move oozing it gave Liverpool the lead. Shaqiri won the ball back in midfield and then played a tremendous pass to Andrew Robertson, who pulled it back for Firmino in the centre of goal to take a touch before drilling it in via a deflection.
Red Star were banned from having any fans in the stadium because of trouble at a previous match in Salzburg, but it was barely noticeable so marginal were they as an attacking force.
The goal settled Liverpool down and it was only a matter of time before they grabbed a second.
Robertson probably should have been the man but he sliced woefully wide in the 43rd minute after finding himself clean through.
Two minutes later the hosts did have their second, Shaqiri again the instigator with a ball through to Salah, who had once more looked rather hesitant but this time acted decisively to slam the ball into the net.
If the contest was still just about alive, it was ended six minutes into the second half when Salah converted from the penalty spot.
Referee Daniel Siebert took an age to decide that Filip Stojkovic’s block on Mane was worthy of a penalty, much to the annoyance of Red Star.
Liverpool continued to push forward, and Firmino drew a decent save from Borjan, but Klopp’s mind began to turn to Saturday’s clash with Cardiff and he withdrew both Shaqiri, who received a standing ovation, and Salah.
Moments later Liverpool were awarded a second penalty for a handball by El Fardou Ben Nabouhane but Mane was denied by a superb save from Borjan and was unable to put away the rebound.
Redemption arrived only four minutes later when he timed his run expertly to collect a Daniel Sturridge pass and poke the ball past Borjan.