The Reds had taken just one point from their previous two games, played away from home last week, so they would have been pleased to take all three this time around as they seek to get closer to the top six of the table again.
For Southampton, the result brought to an end their four-game streak of two wins and two defeats, and they remain in the relegation zone in the league.
Liverpool dominated the first half, but it took until the 42nd minute for them to take the lead thanks to Daniel Agger's header into the top corner. Last season's toughest opponent, the woodwork, also made a return to Anfield as both Jonjo Shelvey and Luis Suarez struck the post and crossbar.
The second half yielded no further goals—here are six things we learned in the game.