Sadio Mane, Andrew Robertson and Salah goals had seemingly put Jurgen Klopp's side in complete control in what was their first European game on home soil since triumphing in last season's competition.
They were left stunned, however, when the visitors responded via goals from Hwang Hee-chan, Takumi Minamino and substitute Erling Haaland.
But Salah's sixth goal of the campaign with 21 minutes remaining got the Reds out of jail as they got up and running in Group E.
The visitors served an early warning via Minamino's long-range shot, yet by the ninth minute the Reds were ahead after Mane drove towards goal down the left before playing a one-two with Roberto Firmino and slotting inside the right post.
Robertson doubled the advantage in the 25th minute, finishing a slick move he had started near the halfway line when he swept home Trent Alexander-Arnold's cross from six yards out.
Liverpool made it three 11 minutes later and had Salzburg goalkeeper Cican Stankovic to thank as he parried Firmino's header into Salah's path and the Egypt forward made no mistake from close range.
Salzburg gave themselves some hope six minutes before the break thanks to a superb individual goal from Hwang, who cut inside Virgil van Dijk before firing home.
The home side made a complacent start to the second half and they were punished when Minamino sent a rasping volley into the ground and beyond Adrian.
That was enough to prompt the introduction of in-form Haaland and he soon had his goal, tapping in from close range for his 15th of the campaign to silence Anfield.
But Salah came to the rescue when he latched onto Firmino's flick-on to have the final say in a pulsating match.
What does it mean? Relief for Reds as they escape unscathed
They might have got the result, but Jurgen Klopp will be far from impressed with his team after seeing them surrender a three-goal lead before Salah spared their blushes.
After losing to Napoli on matchday one, it has been a somewhat unconvincing start to the defence of their European crown - though it should be said the Reds were mesmeric for much of the opening half here, building on their fine domestic form. Qualifying from this group should still be a formality.
Sadio on the Mane against old club
Mane took all of nine minutes to remind Salzburg what they are missing with a stunning goal - his seventh of the season in all competitions, The Senegal international scored 31 goals in 59 league starts for the Austrian side, earning himself a Premier League move to Southampton for just £11.8million. The rest is history, of course, but what a bargain fee that looks now.
Uncharacteristic lapses from Reds
Salzburg might be Europe's goal kings, having scored 55 times in their 12 previous outings this season, but Liverpool would have expected to keep them at bay at fortress Anfield. The likes of Joe Gomez and Fabinho had nervy moments, while even the usually unflappable Virgil van Dijk was troubled at times.
Key Opta facts
- Liverpool have won their last 12 home matches in all competitions, their best winning run at Anfield since an 18-game streak between April and November 1985.
- Salzburg became only the fourth team to score three goals away at Anfield in the UEFA Champions League (also Barcelona, Chelsea and 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.
- Salzburg's two Champions League matches this season have seen 15 goals (9 scored, 6 conceded), more than any other team.
- Andrew Robertson is the first Scotsman to score for Liverpool in European competition since Gary McAllister in the 2001 UEFA Cup final.
A trip to Genk on October 23 in what is the first of back-to-back Champions League matches against the Belgian side awaits Liverpool, who return to Premier League action against Leicester City on Saturday. Salzburg, meanwhile, host Napoli in matchday three.