The hosts were able to include strike sensation Erling Braut Haaland, with the in-demand forward on the bench at Anfield when the two sides met earlier this season.
Jurgen Klopp made three changes to the side who won 3-0 at Bournemouth at the weekend, with Trent Alexander-Arnold, Georginio Wijnaldum and Sadio Mane replacing Joe Gomez, James Milner and Alex Oxlade-Chamberlain.
Virgil van Dijk was forced to make two solid challenges as Salzburg made a fast start before home keeper Cican Stankovic pulled off a fine save to deny Mohamed Salah.
Reds goalkeeper Alisson had to be alert to keep out a Takumi Minamino effort and a follow-up from Hee-Chan Hwang, with the game opening up early on.
Stankovic was back in the action to get down smartly to parry a Sadio Mane shot, with neither Naby Keita nor Alexander-Arnold able to make follow-up shots count.
Haaland tested Alisson with a low shot with 25 minutes played before Keita set up Salah, only for the Egyptian to drag his shot wide when he should have done better.
The visitors then began to exert some control and shortly before half-time Stankovic had to make a crucial intervention to prevent Keita from setting up Roberto Firmino.
Early in the second half, Salah passed up another clear opening, lifting his shot over the crossbar after being played through on Stankovic by Mane.
However, the Reds, who had to bring on Gomez to replace the injured Dejan Lovren, did take the lead just before the hour mark as Mane beat Stankovic to the ball and his lofted pass set up Keita for a simple headed finish.
Within two minutes it was 2-0 as Jordan Henderson found Salah, who made up for his earlier misses by taking the ball around Stankovic before finishing superbly from a tight angle.
That double strike knocked the stuffing out of the hosts, with Salah drawing another save from Stankovic with 15 minutes left to play.
The Salzburg keeper then denied Mane as the visitors were able to see out the closing stages of the game in total comfort.