Liverpool were energetic throughout the first half as Diogo Jota put the Reds 1-0 up inside the opening two minutes, before the Portuguese struck again on the half-hour mark to double the lead. Thiago then added a third with a deflected strike as Liverpool took a commanding 3-0 lead into the break.
The Reds made it four at the start of the second half through Virgil van Dijk, who converted from Trent Alexander-Arnold's corner, but no further goals were forthcoming despite good chances for the Reds and Jota in particular.
It ultimately finished in a comfortable 4-0 win to Liverpool, who now sit one point behind league-leaders Chelsea. Next up for the Reds is the Merseyside derby at Goodison Park on Wednesday night as the Reds look to continue their solid start to the campaign thus far.
Both sides made a lively start to the game and it was the Reds that went ahead inside the opening two minutes as Sadio Mane and Andy Robertson combined on the left, with the Scotsman squaring the ball for Diogo Jota to turn home and make it 1-0.
Saints looked for a quick response as James Ward-Prowse forced Thiago into an error on the edge of his box, with Alisson having to make an important interception to deny an equaliser.
Robertson looked to be in the mood and he was a constant threat down the left as he put in another good ball, this time for Mane, who was unable to keep his header on target.
The same combination resulted in the ball in the back of the net just minutes later as Robertson's set-piece was expertly headed home by Mane, but the linesman correctly flagged for offside as the goal was chalked off.
Mohamed Salah was lively on the wing and he got the better of Mohammed Salisu as he outmuscled the Saints defender and looked to curl an effort inside the far post, but the Egyptian was unable to get his shot on target.
Alisson was called into action again as Armando Broja won the ball from a careless Trent Alexander-Arnold throw-in and ran through on goal, with the Reds goalkeeper having to make an important save at the near post to deny the Albania international.
Liverpool finally doubled their lead just after the half-hour mark as Jordan Henderson and Salah combined, with the latter squaring the ball for Jota to tap in from close-range.
The Portuguese could have bagged a hat-trick as he got on the end of Henderson's knockdown inside the box, but the visitors did well to get bodies in the way and smother his shot.
But it didn't take long for the Reds to make it three, with Thiago picking the ball up at the edge of the box and getting a left-footed strike off which took a deflection off Lyanco and flew into the back of the net.
Liverpool continued to have plenty of joy in attacking areas with nearly every attack producing at least a half-chance for the Reds, but Saints did well to thwart the final finish.
But Ralph Hasenhuttl's side had glimmers of hope themselves, with Alisson called into action again following a good strike by Adam Armstrong, which the Brazilian got down to and turned behind for a corner.
The Reds broke at pace from the ensuing set-piece through Salah, who had Jota and Mane to his right, but the Reds forward was unable to find either of the duo as a big chance went begging.
Liverpool started the second half as they left off from the first as Mane's curling strike was put behind by Alex McCarthy for a corner, which the Reds converted. Alexander-Arnold's cross made its way to a free Virgil van Dijk, who finished with a low effort to score his first goal of the season and make it 4-0.
Jota should have bagged his hat-trick as Robertson produced an excellent inswinging cross to find the forward inside the box, but he was unable to get his shot on target on the stretch.
Takumi Minamino was then introduced for Jota in the final ten minutes, with Klopp opting to keep Mane and Salah on the pitch for the full 90 minutes, with the duo yet to get their names on the scoresheet.
However, it wasn't to be for the duo as it ultimately finished 4-0 on another successful day at the office for Klopp's side, who now sit on 28 points, just one behind league-leaders Chelsea.
Liverpool: Alisson, Alexander-Arnold, Konate, Van Dijk, Robertson, Fabinho, Henderson, Thiago, Salah, Jota, Mane
Southampton: McCarthy, Lyanco, Romeu, Ward-Prowse, Armstrong, Adams, Perraud, Broja, Livramento, Salisu, Bednarek
MOTM: Diogo Jota