Reds forward Sadio Mane began the scoring against his former club, Andy Robertson doubled the advantage at the end of a fine team move, and Salah put the home side 3-0 up after 36 minutes.
The visitors produced a remarkable fightback to level the score by the hour mark - but Salah struck his second goal of the match on 69 minutes to secure an important victory.
Here are five talking points from a dramatic Group E encounter...
Mane shows his class
Mane took just nine minutes to remind former club Salzburg of his goalscoring prowess - and the muted celebration that followed his strike reminded them of his class, too.