Southampton are third from bottom, a point behind 17th-placed Huddersfield Town who won earlier on Sunday.
Karius was called into action again in the 29th minute, denying both Guido Carrillo and James Ward-Prowse.
The Merseyside club are encouraged with a triple-injury boost alongside Lallana as Alberto Moreno and Ragnar Klavan have also been deemed fit enough to feature against the Saints, while Mohamed Salah will be looking to score his 22nd Premier League goal at St Mary's stadium.
The Egyptian has now scored 30 times in all competitions this season since his summer move to Liverpool FC from Serie A giants AS Roma in a £39m deal.
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The two-goal lead was harsh on Southampton going into half time, but it nearly got worse in the opening stages of the second half when Mane's acrobatic effort was blocked and fell at the feet of Salah, although this time the prolific winger scooped his finish over the crossbar.
However, Salah reduced the Tottenham striker's lead at the top of the goal-scoring charts to one goal following his precise finish on the stroke of half-time after Firmino's world-class assist.
Liverpool were able to remain in control of the contest after their bright opening in the second half, though, dominating possession and preventing the home side from building any momentum in their search for a route back into the game.
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.
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.