The hosts took the lead after just 10 minutes when Xherdan Shaqiri's shot deflected off Shane Long and then Wesley Hoedt for a peculiar own goal.
Joel Matip rose above opposing centre-back Jannik Vestergaard to head in Trent Alexander-Arnold's corner and double the advantage.
Liverpool effectively made sure of victory just before the break as Shaqiri's superb free-kick came down off the crossbar and Mohamed Salah turned the ball in from close range.
Saints were not quite as poor as being three down at half-time suggests. However, they largely sat back in the second half, perhaps content not to be embarrassed - though Salah did have a goal disallowed for offside late on.
Jurgen Klopp's side go back to the top of the Premier League with their sixth straight win in the competition, and they also beat Paris St-Germain in their Champions League opener on Tuesday.
Liverpool won eight straight league games and 10 in all competitions from the start the 1990-91 season.
Three and easy
The second half of this game at times felt almost like an extra training session for both sides given how emphatically Liverpool's first-half spree had ended the match as a contest.
Their opener was fortuitous but did originate from Sadio Mane's superb pass to pick out Shaqiri on the left before the Swiss - who was making his first Liverpool start - cut back onto his right and tried to pick out the far corner.
Long blocked the strike but Hoedt could not prevent the deflection rebounding off his thigh and past the scrambling Alex McCarthy.
Mane, Salah and Roberto Firmino - making his full return after an eye injury - scythed through Southampton at times but could not quite find the decisive final touch, with Salah denied by a fine Cedric Soares tackle after a one-two with Firmino.
But if the Reds' poise was slightly off, the power was not - Matip thumping a header into the top of the net from the corner that followed Cedric's intervention.
Shaqiri was then denied by the crossbar after beating McCarthy from 25 yards with a sublime, dipping free-kick, but Salah was sharp enough to get to the rebound first and score his third of the season.
Sterner tests await
Aside from the 4-0 thrashing of West Ham on the opening weekend, this was perhaps the easiest win of Liverpool's record run and Saints' complete indifference to chasing the game in the second half afforded the hosts welcome comfort after the frenetic 3-2 win over PSG.
The only real moment of concern was when Virgil van Dijk, solid in defence whenever his former side did foray forward, had to come off after being struck on an already bruised rib, though Klopp said after the game the injury is not serious.
In a tough run of fixtures, Liverpool next face Chelsea back to back in the Carabao Cup and Premier League, before travelling to Napoli in the Champions League and then hosting champions Manchester City in the league on 7 October.
Those opponents, not an insipid Southampton, will give a better indication of how strong this Reds side are.
Spiritless Saints
Southampton were unlikely ever to try and steamroller Liverpool but they were limited by Danny Ings being unable to play against his parent club and fellow forward Manolo Gabbiadini ruled out with a hamstring problem.
Instead it was left to Long, who has now scored just twice in his past 52 appearances, to lead the line and the Republic of Ireland striker was isolated for long periods.
They did create openings - Nathan Redmond got in behind down the right several times, while Ryan Bertrand picked out Pierre-Emile Hojbjerg unmarked in the area, only for the Denmark midfielder to scuff his effort.
Yet it took until the 91st minute for Saints to have a shot on target when substitute Charlie Austin struck straight at Alisson.
Ultimately, that unlucky own goal and Vestergaard's mistake in allowing Matip to get above him for the second killed any interest Mark Hughes' side had in challenging a clearly superior side.
They sat deeper and deeper as the second half wore on, more reminiscent of a side protecting a 1-0 lead than trying not to concede a fourth - and they almost failed in that endeavour too, but Salah came back from an offside position to turn in James Milner's blocked shot.
Man of the match - Xherdan Shaqiri (Liverpool)
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