Georginio Wijnaldum fired over as Liverpool looked for a third, and then Firmino delayed his cross too long after breaking down the left with options to choose from in the middle.
It was a sloppy start from the home side, and Liverpool nearly doubled their advantage shortly afterwards when Alex Oxlade-Chamberlain - one of five former Southampton players in the visitors' matchday squad including January arrival Virgil van Dijk - saw his deflected drive kept out by Alex McCarthy.More news: Indian PM lays wreath on arch-terrorist Yasser Arafat's grave
Having ended their 12-game winless run with victory at West Bromwich Albion last week, the Saints will be looking to secure back-to-back wins in the Premier League for the first time since April 2017.
Karius was called into action again in the 29th minute, denying both Guido Carrillo and James Ward-Prowse.
Southampton continued to knock on the door, though, and Ward-Prowse was on the other end of two unsafe crosses to the back post but saw both of his headers saved by Karius.
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.More news: Coastline Trust Co Has $4.81 Million Holdings in Kimberly Clark Corp (KMB)
The 26-year-old has now scored 20 goals in all competitions in a single season for the first time since 2013-14 with Hoffenheim.
It was a sleepy start to the second period, with Liverpool under no threat as the Saints showed no sign of being able to find a route back into proceedings.
Firmino nearly extended Liverpool's lead on 71 minutes only for Alex McCarthy to make a brilliant save before Salah failed to convert the rebound.
Ward-Prowse hit a hard free kick to the front post, and Hoedt couldn't quite turn it past Karius as the match remained 2-0 in the 60th minute.More news: Indian PM makes historic visit to Palestine
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