With just 6 minutes on the clock, a long ball forward towards Mohamed Salah saw Wesley Hoedt fluff his attempted clearance, allowing Salah to slide a perfectly-weighted ball across to the incoming Roberto Firmino on the edge of the home side's box.
And while they were not at their best, Klopp was left to reflect on a Liverpool performance that improved throughout.
Liverpool manager Jurgen Klopp has warned Manchester United counterpart Jose Mourinho that he faces a fight to the bitter end in the race for second place after his side chalked up a regulation Premier League win at Southampton.
Saints boss Mauricio Pellegrino, meanwhile, lambasted the attitude of his players during the second half as they failed to build on last week's triumph at fellow strugglers West Brom, meaning his side have won just once in 14 top-flight matches.
Salah's role in the two goals should not be underestimated but Firmino's brilliance was timely given Liverpool legend John Aldridge's quotes earlier in the week.
He also said he has no idea how Firmino had the vision to back heel the ball to set up Salah
Van Dijk began the move as he cut out a cross with his head before Liverpool 'keeper Loris Karius freed another former Southampton player Alex Oxlade Chamberlain. "They will want to show their quality but we have our own targets".
Firmino slotted his ensuing finish home with aplomb, before the Reds' brilliant attacking duo combined once again to double Liverpool's lead shortly before half-time.
"But Virgil is part of the past and we have to focus on our game and what we have to do".
Southampton versus Liverpool games are like a derby nowadays so the fans and players should be well up for this one. I don't think nine points from Arsenal is enough to have rest but it is nine points.
Here's every single new emoji coming to iPhone and Android this year
The finalized list included a softball, a pirate flag, a "woozy face", and, in a major win for Sen. With the release of 157 new emoji's the mango will appear as a part of Unicode's new update.