Jurgen Klopp's men needed to avoid defeat to be sure of continuing their European title defense – something that looked far from assured in a high-octane first half, where there were openings at both ends.
Naby Keita made the 57th-minute breakthrough before Mohamed Salah added a wonderful second 100 seconds later, although goalkeeper Cican Stankovic will rue his part in both goals.
Highly-rated Salzburg striker Erling Haaland was unable to add to his eight Champions League goals this season, but few Europa League opponents will relish facing him and his relentless team-mates when they drop down after the turn of the year.
Stankovic was forced into a save from Salah when the Liverpool forward beautifully brought down Dejan Lovren's raking ball, but Salzburg made much of the running in a frenetic opening.
Hwang Hee-chan and Takumi Minamino's invention and clever movement was causing the Reds problems and Alisson made a brilliant double save in the seventh minute to deny each man from close range.
Haaland's pursuit of a goal in every Champions League group game ran into Alisson in the 24th minute after he sprung onto a Minamino pass behind Lovren.
Salah dragged woefully wide of the near post and Stankovic stood firm when the Egypt star played Keita through in first-half stoppage time.
Haaland blasted a promising chance into the side netting and Salzburg would rue their marksman's unusual wastefulness as they conceded twice in quick succession before the hour.
Stankovic had charged out boldly to check Salah near the edge of his box shortly after the restart but was too ambitious when Sadio Mane rounded him down the left channel and crossed for Keita to head in.
Salah then danced around the over-zealous Austria international but still had plenty to do – the finish from a tight angle on his weaker right foot was sublime.
Roberto Firmino, Salah and Mane all went close to a third goal during the closing stages as Salzburg's exertions took a clear toll.