The match saw Virgil van Dijk make his first return to St Mary's and he received a hostile reception, with Saints fans jeering him whenever he touched the ball. The Egyptian forward laid the ball through to Firmino and the Brazilian finished the one-on-one easily to score his first goal in seven games.
And while they were not at their best, Klopp was left to reflect on a Liverpool performance that improved throughout.
It was by no means a vintage display from Jurgen Klopp's men, but they did ultimately cruise thanks to the individual quality of Firmino and Salah, who have now scored 34 of Liverpool's 61 Premier League goals this term.
Klopp's side could have added further gloss to the scoreline as they missed a couple more opportunities, but Southampton never looked like fighting back and Liverpool successfully capitalised on United's 1-0 defeat at Newcastle United earlier on Sunday.
Nonetheless, Matip is looking forward to improving his game alongside the Dutch defender, and help the team to secure the Champions League promotion.
His pass to Salah was turned inside by the Egyptian for Firmino to apply the finishing touch for his 20th goal of the season.
"We couldn't show what we are about, it was too much for us".
However, they've also been dumped out of the FA Cup at the 4th round stage at home to bottom of the Premier League West Brom and have lost in the league away at 17th placed Swansea (they were bottom at the time).
One-time Southampton player Alex Oxlade-Chamberlain tried his luck moments later, with a deflected strike being stopped and held by Alex McCarthy.