The match saw Virgil van Dijk make his first return to St Mary's and he received a hostile reception, with Saints fans jeering him whenever he touched the ball.
Firmino was the creator on this occasion, though, with the former Hoffenheim star having produced what many have since referred to as the "assist of the season" in setting Salah up.
Southampton responded well and exerted pressure on their visitors, with Pierre-Emile Hojbjerg forcing Karius into action from close range in the 18th minute.
Salah's role in the two goals should not be underestimated but Firmino's brilliance was timely given Liverpool legend John Aldridge's quotes earlier in the week.
Southampton manager Mauricio Pellegrino said: "Liverpool is a huge possibility for us but a really tough side".
Liverpool boss Jurgen Klopp on Southampton explained: "They're a good team, to be honest I don't get why they are in the region of the table they are".
Sword-wielding attacker in Yogyakarta church shot by cops
Lidwina Church in Yogyakarta province, Indonesia, following an attack by a sword-wielding man during Sunday Mass, Feb. 11, 2018. A priest and several churchgoers have been injured during a machete attack on a church in the Indonesian city of Yogyakarta.
Firmino returned the favour from the first goal with an audacious backheel into Salah's path and he slotted the ball into the bottom corner for his 22nd league goal of the campaign to close back to within one of Harry Kane in the golden boot race.
It was satisfying to see attacking intent at the Hawthorns as we managed to score three goals away from home and should've scored more!
Klopp's full pre-Saints press conference.
Salah is proving to be the signing of the season as his goals, influence and general play have given something Liverpool have lacked for some time, and despite the fact he is not an out-and-out striker, he is scoring goals like one.
Southampton: McCarthy, Cédric, Stephens, Hoedt, Carrillo, Tadić, Romeu, Ward-Prowse, Lemina, Bertrand (c), Højbjerg.