Liverpool moved to within two points of a Champions League place by scoring with their first two shots on target at fifth-placed Southampton.
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 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.
Liverpool had already survived a penalty appeal by that point, as on-loan Benfica midfielder Djuricic burst into the area and appeared to be pushed over by Can, but referee Kevin Friend gave nothing.
Southampton were appealing in vain again within a minute of Coutinho’s goal, as Djuricic looked to have been fouled by Allen, with Eljero Elia’s follow-up shot saved by Simon Mignolet amid the confusion.
The visitors had a penalty claim of their own rejected when Jose Fonte took Sterling’s legs from underneath him – although the defender could argue he got the ball first.
There was more controversy before the break, with Mignolet appearing to handle just outside the box as he raced out to stop Elia reaching Nathaniel Clyne’s through ball.
Markovic, despite his assist for Coutinho’s goal, looked uncomfortable in the left wing-back role as Liverpool wobbled defensively, and Rodgers’ decision to replace him at half-time with Alberto Moreno gave the visitors a more solid look.
The hosts failed to open the Reds up as they had done in the first half, and conceded a second when Moreno broke down the left after Morgan Schneiderlin had given the ball away, and Targett’s scuffed clearance fell for Sterling to sweep in.
By then, Southampton had run out of ideas, and Rodgers’ side had a fourth successive victory secured.