Wesley Hoedt's unfortunate own goal spurred the hosts into life, with Joel Matip's excellent header doubling their advantage.
Salah raced up the other end and almost 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.
And while Klopp insists Van Dijk's injury is nothing serious, Matip will not be afraid to step up if required.
Southampton failed to mount a comeback after the break and the result left them languishing in mid-table.
'I was not too lucky with scoring goals during my time at Liverpool but I hope that maybe this season I can score a few more goals.
Relentless Liverpool march on.
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. Although Liverpool dominated the game in its entirety, they made it hard for themselves in order to get full points from that game.
21' - GOAL! Liverpool 2-0 Southampton (Matip). The Reds are cruising now.
Liverpool supporters will be already looking forward to seeing him against Chelsea in the Carabao Cup on Wednesday, with Klopp confirming his half-time change was for tactical reasons rather down to fitness. Despite initial claims from the Swiss playmaker, his dead ball didn't cross the line and it was definitely Salah's goal.
Liverpool have gone 750 minutes without conceding a Premier League goal at Anfield, keeping eight clean sheets in a row.
Hughes feels the current Liverpool squad under Jurgen Klopp have "travelled a long way down the road" in the club's quest for a first domestic championship since 1989/90.
Southampton didn't have a shot on target until the 91st minute.