Things only got worse in the opening stages of the match too as Liverpool broke the deadlock after just six minutes, with Salah pouncing on a mistake from Wesley Hoedt before teeing up Firmino, who buried his confident first-time finish into the bottom corner.
Hosts Southampton have made just one change to the starting XI that won 3-2 at West Bromwich Albion last week, with Pierre-Emile Hojbjerg replacing Sofiane Boufal in midfield.
The two team-mates displayed their telepathic understanding as Firmino, with his back to goal, flicked the ball into Salah's path in defence-splitting one-two.
Salah and Firmino both squandered chances to double their personal tallies in the second half as Southampton took risks and left gaps at the back.
Van Dijk has not had the start he would not wanted to his Liverpool career after being dropped for their game against Huddersfield following a string of underwhelming performances.
The African forward started the weekend one goal adrift of the Premier League's leading goal-scorer Harry Kane before the England worldwide netted the victor in the north London derby. First, Firmino's shot was well saved by Alex McCarthy as Salah dragged the rebound wide, Sadio Mane then hammered a shot wide as he missed the chance to ensure that all three of Liverpool's forwards were on the scoresheet, before Adam Lallana then curled wide after Salah's incisive pass.
It meant ex-Saints defender Virgil van Dijk enjoyed a happy return to the south coast following his £75million move to Liverpool last month.
"I enjoyed it a lot", Van Dijk said.