The Reds played 120 minutes just three days ago as they beat Chelsea on penalties in the Super Cup in Istanbul.
Sadio Mane and Roberto Firmino both found the back of the net as the visitors controlled possession throughout the match.
Ex-Liverpool striker Danny Ings grabbed a goal back in the final 10 minutes, but the Merseysider’s held firm to pick up all three points.
And speaking after the game, Klopp revealed his delight at the “respect” his men showed to the match, despite the quick turnaround from Wednesday.
“We are not used to winning big things and then coming to Southampton, I really loved how much respect we showed today,” the German said.
“I said before the game that this looks like the biggest banana skin in history; everybody is waiting for it and probably all the headlines were written already.
“I told the boys the headline I would prefer is ‘the mentality giants were in town’ – that’s how I feel it in the moment.
Latest Liverpool news
“The boys did an outstanding job for today, all good, let’s go home, recover and start again.”
Klopp admits that his side looked tired towards the end of the match, but he was pleased with the overall performance on the south coast.
“We controlled the game, we were fluent, all that stuff and then with the long balls and set-pieces Southampton came up then,” he continued.
“But Sadio scored a fantastic goal in pretty much the best moment at least of the first half, in the last second pretty much.
“That helped, of course, and in the second half we made other big steps.
“I think then the boys felt the tiredness, that was the moment you think, ‘Really? Now we have to go again?’
“So, we kept the result, I think we deserved the three points and I’m completely fine with that.”
- SUCCESS STORY Bidz.com Auctions: Consumers Love to Win
- Overwatch Uprising Infographic Reveals A Surprising Win/Loss Ratio
- Idea Factory International Reveals Lovely Limited Edition for Superdimension Neptune VS Sega Hard Girls
- French Nintendo figures reveal love for red Wii
- E-BUSINESS SPECIAL REPORT Who Will Win the Mother's Day Race?
- Simon Singh Wins Maiden Leelavati Award
- The 2013 E3 Epic Win Awards (part 4)
- Fossilized Pollen Reveals History of Northern Antarctica
- Call of Duty: Black Ops – multiplayer gameplay impressions from LA reveal
- Indie kings – The Pickfords win with Magnetic Billiards