Jurgen Klopp's men were in full control in the first half, and they managed to bag a third goal just before the break when Salah poked home from close range after Xherdan Shaqiri's free-kick hit the woodwork.
Cedric Soares makes a superb last-ditch tackle to deny Mohamed Salah, but from the resulting corner Joel Matip rises to thump in Liverpool's second.
LIVERPOOL, ENGLAND - SEPTEMBER 22: Wesley Hoedt of Southampton scores an own goal past Alex McCarthy of Southampton for Liverpool's first goal during the Premier League match between Liverpool FC and Southampton FC at Anfield on September 22, 2018 in Liverpool, United Kingdom.
Scoring against Southampton will have come as a huge relief and after the game he showed his class by making one young Liverpool fan's dreams come true.
"It was not easy, to be honest".
'It's not cool, but it should not be too serious. It's my job to sometimes deliver news that nobody wants to hear.
He continued: "Today we had a few problems but we can still increase".
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"It's just good to have him".
'He should be absolutely okay, hopefully. "I made different decisions but his time will come, that's for sure".
At the other end of the table, Burnley hauled themselves off the bottom with their first win of the season, 4-0 over Bournemouth to interrupt the Cherries' fine campaign, with goals from Matej Vydra, Aaron Lennon and a double from Ashley Barnes.
Liverpool and Chelsea are among the only two sides to still have a 100 per cent record in the Premier League and the Merseyside outfit will seek to extend their brilliant start against an inconsistent Southampton side.
"It's a surprise to win seven games because of course you can not expect it, but all performances we showed so far were at least good enough to win the games", stated Klopp.
"I have no clue yet what I will say in the team meeting but I don't think that putting more pressure on, saying it will be historical or whatever would help".
However, Salah - in need of a goal to quell the largely ridiculous talk about his form - reacted quickest to convert.