The Reds maintained their 100 per cent start to the new season with a narrow victory at St Mary's on Saturday afternoon , thanks to goals from Sadio Mane and Roberto Firmino.
The former opened the scoring right on half-time with a superb curling effort, before the Brazilian doubled the visitors' advantage with a brilliant individual effort after 71 minutes.
Liverpool were forced to play their third game in eight days against the Saints, with little time to prepare following a UEFA Super Cup final victory over Chelsea in Istanbul last Wednesday.
Klopp's men were not at their fluid best against Ralph Hasenhuttl's side, who set up a nervous finale when former Anfield striker Danny Ings capitalised on an Adrian error to score with nine minutes remaining.
The Reds held on for a hard-earned three points and their head coach saved praise for his talismanic forwards who were a "constant threat" once again in the final third.
"A lot of times in my life I had sensational players always [but] not world-class players," Klopp told Liverpool's official website .
"Now, thank God I have them and they make the difference. We don't have to talk about that.
"Sadio Mane, Bobby Firmino, Mo Salah – these guys in the last situation are really outstanding.
"They were a constant threat, however tired they are. That's brilliant and, of course, made the difference. We don't have to talk about that."
Klopp went on to reveal that he made a slight tactical tweak at the break against Southampton, in order to exploit the spaces the hosts were leaving open in wider areas.
"I think Southampton tried to kill our game with their formation and they played differently against Burnley obviously," the 52-year-old added.
"They had a real proper fight at the back, so that gave us space on the wings. We used that but not often enough, otherwise, they would have had to run much more. We could adjust that a little bit at half-time.
"I'm really happy. It's so important. The start of the season is so important. Having an away game at Southampton is always difficult, but in the week we had, it was special."
Next up for Liverpool is a huge showdown against Arsenal, who have also won their first two matches of the 2019-20 campaign, at Anfield on August 24.
The Reds are top of the Premier League on goal difference ahead of the Gunners at the moment, while champions Manchester City are down in third after a 2-2 draw with Tottenham over the weekend .