Dominik Szoboszlai curled in the Premier League leaders' opener on 30 minutes following some dreadful play from the Saints.
Southampton goalkeeper Alex McCarthy played a ball to Mateus Fernandes but the midfielder quickly lost possession in his own box.
It fell to Flynn Downes who tried to clear his lines although his effort fell into the path of Szoboszlai, who swiftly made it 1-0.
Speaking about the defending on talkSPORT, former Arsenal man Perry Groves said: "It was so basically bad it was ridiculous."
As a result, Southampton have made the most errors leading to opposition goals than any other Premier League side this season with seven.
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Liverpool's goal marked a poor start for the Saints, who have already lost nine of their opening league games.
Only seven sides in English top-flight history have reached ten losses in 12 or fewer matches from the start of a campaign and all seven of those have been relegated.
However, Southampton hit back on the approach to half-time when Stuart Armstrong netted the rebound from his saved penalty.
Referee Samuel Barrott awarded the spot-kick after Andy Robertson was adjudged to have fouled Tyler Dibling.
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The decision was reviewed by VAR and, although it appeared as though the contact was made outside of the box, the decision stood.
"The referee's call of penalty for the challenge by Robertson on Dibling is checked and confirmed by VAR, who deemed there was no conclusive evidence that the contact occurred outside the penalty area," said the Premier League Match Centre on X.
Liverpool dominated the first half by enjoying 67.3 per cent of the possession and having 11 shots, with seven of those on target.
However, they were unable to sustain their lead ahead of the break and Southampton took the lead through Fernandes in the second half.
Mohamed Salah levelled the scoring soon after though, with the hosts displaying poor defending once again.
Salah then a late winner from the penalty spot to make it 3-2 to Liverpool following continuous pressure from the Reds.
Victory has taken them eight points clear at the top of the Premier League ahead of reigning champions Manchester City.