The hosts scored three first-half goals to subdue Saints into just limiting the damage after the break and Reds boss Jurgen Klopp said the challenges will get “tougher and tougher” for his side.
“I wasn’t interested in the history before the game but now I like it,” the German told BBC Sport.
“We will try to carry on that path but it’s unbelievable what is coming up now.”
In their next four matches, Liverpool face Chelsea twice, travel to Napoli and then host Premier League champions Manchester City.
“The boys are in good shape. That helps – let’s see if we can carry on,” added Klopp.
On Saturday, the hosts took the lead after 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 added a third 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 sat back in the second half, perhaps content not to be embarrassed – though Salah did have a goal disallowed for offside late on.
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.
The second half of this game felt 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.
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 in the torso, though Klopp said after the game the injury is not serious.
“Before the PSG game, he already had kind of a bruised rib, which is very uncomfortable, and he got another hit there,” said the German.
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.
