The Egypt international scored twice in 10 minutes before half-time, and Philippe Coutinho tucked away a rebound on 68 minutes to round off a comfortable win for the Reds. Salah's eighth and ninth goals of the season mean he's also now the Premier League's leading scorer.
Southampton's second successive league defeat—and their third without a win—leaves them sat four points above the relegation zone, while Liverpool move above Arsenal and into fifth by virtue of goal difference.
Ragnar Klavan partnered Dejan Lovren in the middle of Liverpool's defence with Joel Matip injured, while Trent Alexander-Arnold made his fourth start of the Premier League campaign at right-back.
Shane Long made his first Southampton start in almost a month, meanwhile, and Dutch centre-back Wesley Hoedt came in alongside compatriot Virgil van Dijk at the core of the Saints defence.
Both Van Dijk and Hoedt were forced into early action after the home side showed quick intent to get on the scoresheet, with Coutinho and Salah unsurprisingly looking like the most likely sources of a Liverpool strike.
The Saints were limp for the opening exchanges—a theme that ran for the majority of the fixture—although football writer Chris Williams warned of the missed opportunities running against hosts Liverpool:
Fortunately for the Reds, Dusan Tadic was on hand to help Liverpool break the deadlock after losing the ball on the edge of his own box under heavy pressure, a mistake that Salah took full advantage of.
Some neat passing by the Reds eventually found Salah on the right corner of the box, and ESPN's James Tyler sang his praises after wrapping a sumptuously curled strike into the top-left corner with his left foot:
The writing had been on the wall for Southampton, and Liverpool manager Jurgen Klopp was pleased to see his men open some daylight after two of his elite stars combined to craft a second 10 minutes later.
All credit was owed to Coutinho for his searching through ball to Salah up top, and Anfield HQ's Oliver Bond highlighted the latter's storming start to life back in the Premier League, hinting his £35 million price to be of great value:
Whether his error in the lead-up to Salah's first goal had anything to do with, Tadic was hooked by Southampton boss Mauricio Pellegrino at the break, with striker Charlie Austin the man who replaced him.
But even the addition of another frontman—with Long shifting toward the right flank—had little positive impact in terms of Southampton's press in the Liverpool half, with some players looking as though full-time couldn't come soon enough.
Any such feelings of gloom were only exacerbated when Coutinho seared down on the Saints area to power in a routine finish from a Roberto Firmino rebound.
BT Sport Score pundit Kelly Smith was quick to praise Sadio Mane, however, for his role creating the chance with a run down the right wing:
The remainder of the half played out as Liverpool would have wanted, although Klopp and his men could afford to feel regret they weren't able to add to their three-goal haul.
Nevertheless, three was enough to take the Reds into fifth above Arsenal, who defeated Tottenham Hotspur 2-0 in the north London derby earlier on Saturday.
Liverpool have endured some inconsistent form early on in the 2017-18 campaign, but with their attacking array in such fine fettle, this could serve as just a hint of what's to come from the Merseysiders moving forward.