Danny Ings’ second-minute goal proved decisive at St Mary’s Stadium, with the Reds unable to muster an equaliser against a well-organised, stubborn Saints defence.
Following a below-par first-half display, there was greater intent and tempo about the champions’ play after the interval.
However, Southampton’s defensive perseverance restricted them to a sole shot on target - a scuffed Sadio Mane attempt 15 minutes from time.
Jürgen Klopp’s side remain top of the Premier League on goal difference, although they have played one game more than Manchester United.
Team news
Thiago Alcantara, Alex Oxlade-Chamberlain and Georginio Wijnaldum came in as Klopp revamped his midfield three.