Liverpool's recent raids of Southampton mean that there are a number of sub-plots in this fixture, but the main one from the hosts' side sees manager Mauricio Pellegrino come up against one of his former clubs.
The win sends Liverpool up to third, with the Reds leapfrogging Tottenham.
In terms of form, Liverpool have had a mixed start to the new year with league wins away at Burnley and Huddersfield as well as becoming the first side to beat runaway league leaders Manchester City domestically this season.
But they could have equalised when a deep cross fell to Pierre-Emile Hojbjerg, who got the better of Andrew Robertson but saw his shot blocked by keeper Loris Karius, who was swiftly off his line.
Karius then tipped over a Ward-Prowse header as the Saints found some momentum, and the same player had another chance after Bertrand had crossed with five minutes of the opening half remaining.
It was Liverpool who added the game's second goal against the run of play, though, adding a late first-half strike to their early one as Salah and Firmino once again combined to devastating effect.
The two team-mates displayed their telepathic understanding as Firmino, with his back to goal, flicked the ball into Salah's path in defence-splitting one-two.
Well John, Firmino has scored 20 goals with plenty of games still to play in the league and Europe.
The Dutch defender completely missed his kick when trying to deal with a long ball and Salah pounced in a flash, showing great awareness to square to Firmino, who confidently slotted past Alex McCarthy.