Liverpool's Egyptian midfielder Mohamed Salah (R) celebrates with Liverpool's Brazilian midfielder Roberto Firmino after scoring their fourth goal during the UEFA Champions league Group E football match between Liverpool and Salzburg at Anfield in Liverpo
However, Hee-Chan Hwang pulled one back just prior to the interval, before Takumi Minamino and substitute Erling Braut Håland took advantage of some woeful defending to stun the holders and pull the visitors level on the hour mark.
Erling Haaland of Salzburg celebrates after scoring their 3rd goal during the UEFA Champions League group E match between Liverpool FC and RB Salzburg at Anfield on October 2, 2019 in Liverpool, United Kingdom.
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Jurgen Klopp's men were visibly rattled and were far from their best in the second half, but recovered to snatch the win courtesy of Salah's 12th goal in 13 Champions League appearances at Anfield.
It was a fitting spectacle to mark Liverpool's 100th European Cup match at their famous stadium and saw the holders stretch their unbeaten run on home turf in the competition to 23 games.
The result means the Merseysiders are now level with second-placed Salzburg on three points, with Napoli leading the way in Group E on four. Genk are bottom of the pool with one point.
Next up, Klopp's Premier League leaders face third-placed Leicester City as they bid to make it eight league wins from eight at Anfield on Saturday.
Anfield is always good value on a European night. Liverpool just about put their Matchday One defeat at Napoli behind them with a rollercoaster win against a fearless Salzburg side, who provided the surprise test many were suggesting they might. The Reds played some scintillating football early on, but then showed why some supporters are growing concerned by instances of sloppy defending so far this term.
Salzburg got one goal back just before the break with the Anfield scoreboard suffering a glitch that stated it was 3-3 - or was it simply a premonition? By the hour mark it was all square and Liverpool's swagger was settling into a panic. Kopites are almost used to seeing great comebacks from their team, but not so much from the opposition. In the end, the Merseysiders got the job done and remain a major force on home turf.
Andrew Robertson of Liverpool scores and celebrates during the UEFA Champions League group E match between Liverpool FC and RB Salzburg at Anfield on October 02, 2019 in Liverpool, United Kingdom.
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Klopp's side relish these type of nights under the floodlights and this is evidenced by the fact the German has not lost any of the 20 European matches he has taken charge of in this stadium. This won't go down as one of the great European nights for the Reds, but it was certainly a fun-filled goal-fest to mark a century of European Cup matches in L4.
MAN OF THE MATCH
Mo Salah (Liverpool). Saved his team's blushes with a crucial winner to continue his sensational record in this competition on home turf and claim the headlines with a two-goal salvo.
9' - GOAL! Liverpool 1-0 Salzburg. Mane coolly tucks the ball into the far corner after a lovely one-two with Firmino down the left.
25' - GOAL! Liverpool 2-0 Salzburg. Robertson starts and finishes a sublime move that sees right-back, TAA cross for the Reds' left back to tuck home a rare goal.
36' - GOAL! Liverpool 3-0 Salzburg. The Reds are flying. Salah fires in a rebound after the keeper could only parry Firmino's header into the Egyptian's path.
39' - GOAL! Liverpool 3-1 Salzburg. The visitors pull one back. Hee-Chan Hwang turns van Dijk brilliantly on the left of the box and fizzes a fine finish into the far corner.
56' - GOAL! Liverpool 3-2 Salzburg. The Austrian's are back in it! Hwang escapes down the left and crosses neatly for Takumi Minamino to volley home.
59' - SALZBURG CHANCE. It should be 3-3! Hwang turns brilliantly outside the box and plays in Szoboszlai, but he takes too long pulling the trigger and is eventually rushed into a shot that hits the side netting.
60' - GOAL! Liverpool 3-3 Salzburg. It is 3-3 now! Haaland is left completely free to tap home from close range after neat play from Minamino on the right.
69' - GOAL! Liverpool 4-3 Salzburg. The hosts regain the lead as Salah lashes Firmino's flick-on beyond the keeper.
FC Salzburg are without a win in five matches in all competitions against English teams.
Andrew Robertson is the first Scotsman to score for Liverpool in UEFA European competition since Gary McAllister scored against Alavés in the 2001 UEFA Cup final.
Liverpool conceded three goals in a home Champions League game for the first time since October 2014, when they lost 3-0 to Real Madrid.
Since the start of the 2017-18 season, Roberto Firmino is the only player to have both scored (14) and assisted (10) at least 10 goals in the Champions League; assisting two goals tonight.