Liverpool - who had Virgil van Dijk making his first return to St Mary's Stadium since his £75million move - looked rather more switched on at the back in the second half and Southampton's chances dried up.
Even so, Southampton are unbeaten in their last four league matches, so they have perhaps turned the corner in their quest for Premier League survival.
Goals from Roberto Firmino and Mohamed Salah gave them a comfortable win at St Mary's, but just a week after beating Manchester City in January, Liverpool were back to losing ways and Jamie Carragher thought the loss was typical with a larger problem they have.
A win for Liverpool today will make Klopp the 45th different Premier League manager to record 50 wins.
Firmino returned the favour from the first goal with an audacious backheel into Salah's path and he slotted the ball into the bottom corner for his 22nd league goal of the campaign to close back to within one of Harry Kane in the golden boot race.
"We knew the strengths of Southampton and how they want to play football, so we wanted to defend and change the game to counter", Klopp told Sky Sports.
"Last season, little by little, Jack was playing more in the second part of the season and that is more or less the same now".
Mauricio Pellegrino's first six months in the Southampton job were similar to his short stint as a centre-half at Liverpool: underwhelming. "The only thing we can do is win our games, but it will stay tight until the end".
Karius, easily the busier of the two goalkeepers, was tested again just before the half-hour mark, halting Guido Carrillo's header from a James Ward-Prowse cross.