However, Sadio Mane netted a superb goal against the run of play late in the first half, with Klopp's side then dominating possession after the break.
Roberto Firmino scored an excellent goal with a powerful low shot from inside the box, doubling Liverpool's lead as they looked on course for victory.
A blunder by goalkeeper Adrian saw Danny Ings give the hosts late hope, however, the Reds held on for all three points.
Speaking to reporters after the game, Klopp said: “I said before the game this looks like the biggest banana skin in history. The headlines were already written.
“The headline I would prefer is ‘Mentality Giants Were in Town'. The boys did outstandingly well today.
“We controlled the game, we were fluent, all that stuff and then with the long balls and set-pieces Southampton came up then.
“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.
“We really controlled the game better, did a lot of good things, and then minute 83, I saw it up on the screen, and Adrian thought ‘I played really well, so maybe I have to make the same mistake as Ali did!' and kept the game a little bit open.
“So then they came up and Ingsy had a big chance. That's normal and 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.”
Klopp was keen to defend Adrian's error for Saints' consolation goal, citing Alisson's similar mistakes in the first few weeks of last season to deflect blame away from the Spaniard.
“[I told him] welcome. Ali did the same, obviously it's a goalie thing at Liverpool,” he said.
“No problem with that as long as we win the games.”