The defending European champions boasted one change to the side that won 1-0 at Sheffield United in the Premier League as Joe Gomez comes in for Joel Matip at the back, Red Bull Salzburg’s in-form striker Erling Haaland had to settle for a place on the bench after he missed the Austrian side’s game at the weekend with illness.
Salzburg started brightly but Liverpool almost got their noses in front on eight minutes when Trent Alexander-Arnold got behind the defence and looked to tee up Roberto Firmino, but Jerome Onguene got enough on the ball to take it away from the Brazilian.
It was 1-0 a minute later when Sadio Mane struck against his former club as he exchanged passes with Firmino and coolly slotted his seventh goal of the season past Cican Stankovic.
Liverpool continued to look comfortable with their lead and they eventually extended it on 25 minutes when Andrew Robertson started a flowing move with a surging run, before running towards the near post where he met Alexander-Arnold’s subsequent cross and finished with aplomb.
Salzburg went close to pulling a goal back just after the half-hour when Trent-Alexander played Takumi Minamino onside as he collected a pass from Dominik Szoboszlai and forced a decent save from Adrian at his front post.
However, it was 3-0 to the Reds on 36 minutes when Mane raced away down the left and picked out Firmino in the middle. Stankovic managed to parry his effort but the rebound rolled out to Mohamed Salah six yards from goal and he made no mistake.
Salzburg did manage to reduce the deficit three minutes later when Hee-Chan Hwang managed to cut inside past Virgil van Dijk and blasted a strike beyond the dive of Adrian.
Adrian almost gifted Salzburg a second early in the second half when Patson Daka chased down his attempted clearance the ball rebounded off him and just wide of the goal.
Hwang then fired wide as Salzburg continued to press, while Mane was off-target after meeting a Robertson cross in the centre of the area on 54 minutes.
It was game-on two minutes later when Hwang got in behind Alexander-Arnold down the left and he picked out Minamino who smashed an impressive volley beyond Adrian.
Szoboszlai should have equalised moments later, but Salzburg completed their remarkable comeback on 61 minutes when Minamino tuned provider from the right as he teed up Haaland to tap-in the equaliser with his first touch after stepping off the bench.
Liverpool did turn the tide back in their favour and they managed to get back in front on 69 minutes when a clever flick from Firmino played in Salah and the Egyptian lifted his effort over Stankovic.
Salzburg pushed the Reds to try and find another equaliser, but Jurgen Klopp’s men held on to join their opponents on three points in the group, while Napoli are a point better off at the top.