The winger struck during the first half but Southampton encountered little problem in seeing out the win against Jurgen Klopp's side, who rarely threatened throughout the 90 minutes at St Mary's.
However, Southampton soon rallied and while Loris Karius did well to prevent Nathan Redmond from finding the back of the net, he was powerless to do so two minutes later after a mistake from Ragnar Klavan.
The Liverpool centre-back failed to deal with a ball around 30 yards from goal and Jay Rodriguez took advantage to set up Redmond, who found the bottom corner of the net from 10 yards out.
Klopp would have expected a response from his side but they struggled to put anything together in the final third, and Southampton ended the first half unfortunate not to double their lead.
Dusan Tadic somehow managed to find Redmond at the back post but despite making a strong connection with his first-time strike from eight yards, Karius pulled off an excellent intervention to keep his team in the match.
Southampton returned for the restart on top, but they were unable to create any more opportunities and it gave Liverpool the platform to start to look for an equaliser.
However, aside from two tame efforts from Adam Lallana and Firmino, the Merseyside giants continued to offer very little in the final third and Southampton should have taken advantage midway through the second half.
A rapid breakaway involving Redmond led to the winger playing in Cedric Soares down the right-hand side of the penalty area but rather than cross for Rodriguez, the full-back fired into the side-netting from 12 yards out.
Chances remained few and far between but with seven minutes remaining, Southampton came within millimetres of adding a second through Redmond, who clipped the ball onto the underside of the crossbar from close range after being found by Shane Long.
Philippe Coutinho had made his return from an ankle injury shortly after the hour mark, but he capped an ineffective showing with a 20-yard shot which went into the stands during added-on time as Southampton endured few difficulties in ensuring that they take an advantage to Anfield in two weeks' time.