Philippe Coutinho scored with a 25-yard curler in the third minute.
Southampton had two early penalty appeals turned down, as challenges on Filip Djuricic by Emre Can and Joe Allen went unpunished.
Raheem Sterling also had a penalty claim rejected, but made it 2-0 when Matt Targett failed to clear a cross.
Liverpool are the only team not to have lost a Premier League match in 2015 and are well placed to push for a top-four finish in their final 12 matches.
The result saw Brendan Rodgers' side leapfrog Tottenham into sixth and left them three points behind third-placed Arsenal.
They had to ride their luck, though, to hold on to the early lead given to them in spectacular fashion by Coutinho.
The Brazil midfielder gathered Lazar Markovic's pass and hit a third-minute strike that bounced in off the underside of the crossbar. abd71
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