The Reds were two up inside 21 minutes and made the result safe as early as half time when Mohamed Salah ended a four-game scoring drought on the stroke of the break.
It is the first time in Liverpool's history that they have begun a campaign with seven successive wins in all competitions, following Tuesday's late Champions League triumph over Paris Saint-Germain.
Jurgen Klopp drafted in Xherdan Shaqiri for his full Liverpool debut and the former Stoke City winger played a role in the opening goal on 10 minutes, cutting inside onto his right foot for a shot which Wesley Hoedt clumsily steered into his own net.
Southampton responded reasonably well as Jannik Vestergaard went close from a corner, and Virgil van Dijk, playing against his former club, had to be expertly positioned to ensure that Shane Long did not make the most of some good crosses into the box.
The visitors soon slipped further behind though, and it was another player to benefit from Klopp's rotation who doubled Liverpool's lead as Joel Matip got a run on 6ft 6in Vestergaard to head his first goal at Anfield beyond Alex McCarthy and into the top corner.
Klopp's men surrendered a two-goal lead to PSG in midweek but there appeared to be little danger of a repeat, with Pierre-Emile Hojbjerg passing up one of Southampton's few chances when he completely scuffed his half-volley from Ryan Bertrand's cross.
Salah raced up the other end and nearly scored within 15 seconds of Hojbjerg's miss, back-heeling wide of the post from inside the six-yard box, while Sadio Mane tested the handling of McCarthy as Liverpool knocked on the door for a third before the interval.
The Saints could not get to the break for some respite as Shaqiri, who was surprisingly hooked at half time despite his contribution to two goals, whipped a 30-yard free kick against the crossbar, and Salah raced in to tap home the rebound from only a yard out.
Liverpool strolled through much of the second period in second gear, although the loss of Van Dijk to injury may have soured Klopp's afternoon ahead of next week's double header against unbeaten Chelsea, who visit Anfield on Wednesday in the EFL Cup before welcoming the Reds to Stamford Bridge on Saturday.
Joe Gomez, rested from the start today, deputised with authority as the Reds made it eight top-flight home matches without conceding.
It was not until the final 10 minutes that Liverpool looked like adding to the score, as Andrew Robertson volleyed narrowly over the crossbar from 25 yards and Salah saw a goal disallowed for offside during a goalmouth scramble in which James Milner had two efforts blocked.
With a sixth win from six, Liverpool go clear at the top of the table although Chelsea can join them on 18 points with victory over West Ham United tomorrow.
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold, Matip, Van Dijk (Gomez, 55'), Robertson; Wijnaldum (Keita, 70'), Henderson, Shaqiri (Milner, 46'); Mane, Salah, Firmino
SOUTHAMPTON (4-5-1): McCarthy; Cedric, Vestergaard, Hoedt, Bertrand; Redmond, Hojbjerg, Lemina, Romeu (Bednarek, 55'), Targett (Armstrong, 55'); Long (Austin, 79') body check tags ::