Premier League: Southampton 0 Liverpool 2
Philippe Coutinho scored a magnificent long-range opener amid a frantic start when the home team had a pair of strong penalty appeals turned down.
Southampton enjoyed sustained periods of pressure, but Liverpool's defence managed to hold firm, leaving Raheem Sterling to double their advantage in the 73rd minute.
The result moves Liverpool, unbeaten in the league since mid-December, up to sixth in the table - a point behind their opponents and two shy of Manchester United in fourth.
Filip Djuricic was handed a first start in attack for Southampton, who welcomed back left-back Matt Targett and midfielder James Ward-Prowse.
A hip injury to Liverpool defender Mamadou Sakho meant Dejan Lovren joined Adam Lallana in starting against his former club, while Sterling replaced Daniel Sturridge.
Djuricic was immediately into the action, going down under pressure from Emre Can in the penalty area after 20 seconds, but referee Kevin Friend was unmoved.
Liverpool responded to that uncertain opening by stylishly grabbing a third-minute lead when Lazar Markovic fed the ball to Coutinho, who netted superbly from 30 yards via the underside of the crossbar.
There was more frustration for Southampton as another challenge on Djuricic inside the box, this time from Joe Allen, went unpunished and Simon Mignolet made a fine save when the loose ball broke to Eljero Elia.
Friend provoked further ire from the home support two minutes before half-time when Mignolet charged down Elia's effort on the end of a long punt forward from marginally outside his area and was not punished for handball.
The Belgium goalkeeper then beat away a Djuricic strike to ensure Liverpool's advantage remained intact at half-time.
Having seen his defence under sustained pressure, Liverpool boss Brendan Rodgers sent on Alberto Moreno in place of Markovic at left wing-back, while Morgan Schneiderlin replaced Steven Davis for Southampton.
Graziano Pelle scuffed wide from 20 yards following a slick passing move as the hosts remained on the front foot.
Southampton manager Ronald Koeman introduced Sadio Mane in place of Ward-Prowse and the forward had a shot on the turn blocked by Martin Skrtel as the hour approached.
Sturridge bolstered the Liverpool attack in place of Lallana and some wonderful skill from the England man beat Jose Fonte before he failed to get a shot away.
Rodgers' men sealed the points when Targett lost his footing in trying to clear Moreno's low cross and Fraser Forster could only touch Sterling's firm low strike into the net.
In the closing minutes, Sturridge twice came close to getting his name on the scoresheet as Southampton absorbed their latest setback in a run of one win in four league outings - scoring a solitary goal during this time.