It was by no means a vintage display from Jurgen Klopp's men, but they did ultimately cruise thanks to the individual quality of Firmino and Salah, who have now scored 34 of Liverpool's 61 Premier League goals this term.
Any hopes Mauricio Pellegrino had of his side keeping things tight early on vanished after just six minutes, as Liverpool punished a Wesley Hoedt error to open the scoring through Firmino – the alert Salah setting him up.
Southampton were the better side for the remainder of the first half, but visiting goalkeeper Loris Karius ensured they failed to make the most of their superiority and a glorious exchange in which Firmino repaid the favour to Salah in the 42nd minute saw Liverpool double the lead against the run of play.
29 – Mohamed Salah has been directly involved in 29 @premierleague goals this season (22 goals, 7 assists); the most by a @LFC player in their debut season for the club in the competition. Heroic. pic.twitter.com/4YNjRW3Xqg — OptaJoe (@OptaJoe) February 11, 2018
Liverpool – who had Virgil van Dijk making his first return to St Mary's Stadium since his £75million move – looked rather more switched on at the back in the second half and Southampton's chances dried up.
Klopp's side could have added further gloss to the scoreline as they missed a couple more opportunities, but Southampton never looked like fighting back and Liverpool successfully capitalised on United's 1-0 defeat at Newcastle United earlier on Sunday.
One mistake was all it took for Southampton to find themselves behind early on, with Hoedt culpable.
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.
Southampton responded well and exerted pressure on their visitors, with Pierre-Emile Hojbjerg forcing Karius into action from close range in the 18th minute.
Karius, easily the busier of the two goalkeepers, was tested again just before the half-hour mark, halting Guido Carrillo's header from a James Ward-Prowse cross.
And the German did brilliantly a few moments later, tipping Ward-Prowse's header just over.
Southampton were ultimately made to rue their missed chances when Salah finished with aplomb after a brilliant one-two with Firmino, who set the Egyptian up with a delightful backheel just before the break.
Salah afforded Saints a let-off early in the second half, shooting over from just inside the area after a Sadio Mane effort ricocheted right into his path.
The hosts were rather less effective going forward in the second period and their only chance fell to Hoedt in the 59th minute, but he could only find the side-netting when flicking on a Ward-Prowse corner.
Liverpool should have put the game beyond Southampton 20 minutes from time when Firmino saw his effort saved by McCarthy and Salah put the rebound into the side-netting.
Mane and substitute Adam Lallana had further opportunities in the final 15 minutes, though neither miss mattered a great deal as Liverpool clinched all three points with minimal fuss.
Key Opta stats:
- Jurgen Klopp has won 50 of his 95 Premier League games in charge (D28 L17) - the ninth fastest manager to do so in terms of games (95).
- Indeed, Klopp’s predecessor Brendan Rodgers had also won exactly 50 of his first 95 Premier League games in charge of the Reds (D23 L22).
- Southampton have won just once in their last 14 Premier League games (D7 L6), keeping just one clean sheet in the process.
- Liverpool have scored more Premier League away goals in this campaign (34) than they managed in the entirety of 2016-17 (33).
- Mohamed Salah has been directly involved in 29 Premier League goals this season (22 goals, 7 assists), the most by a Liverpool player in their debut season for the club in the competition.
- Salah has scored 19 left-footed goals in the Premier League this term – the joint-most by a player in a single season, alongside Robbie Fowler for Liverpool in 1994-95.
- Roberto Firmino has scored 20 goals in all competitions in a single season for the first time since 2013-14 with Hoffenheim (22).
- Firmino has scored and assisted in 12 Premier League games, more than any other player since the start of the 2015-16 season.