The Reds travelled to St Mary's with Adam Lallana absent through a groin injury and Philippe Coutinho's involvement in doubt after only returning to the club's training ground on Thursday and not taking part in training.
Though the in-form Brazilian made the starting line-up, Liverpool took time to get into their stride - Sadio Mane letting two good opportunities to force the breakthrough slip before half-time.
Klopp's charges rallied after the break and thoroughly asserted their dominance, but couldn't turn their chances into goals - Roberto Firmino and Nathaniel Clyne guilty of wasting the most clear-cut of those.
Yet the point on the South Coast, according to Klopp, came with a performance that was of higher intensity and quality than he imagined.
Klopp: Liverpool produced a 'really good performance' against Southampton
Asked for his summary by journalists afterwards, the German declared that his team were "not bad first-half" and "much better second-half", adding: "It was a really good performance against a difficult-to-play side. Southampton are one of the best organised teams in the league."
Klopp felt that the Saints "adapted their style a little bit" to counter his team's strengths but insisted they still had "enough chances to win the game" and "especially in the second-half."
He commented that his side's "reaction" after the break was "really good" and that their body language was "better" after a first-half in which it was obvious from Liverpool's display that it was a "really difficult" game.
Klopp declared that his team were "prepared for a difficult game" but said that it "feels differently" in the game. Yet he still insisted he was "absolutely fine with the performance" and that they "could have" and "maybe should have" scored.
"But that's how football is," he continued, insisting that in his managerial career he has "lost games like these" even when his teams "were so much better and had many more chances."
We played more to our usual style than Southampton to theirs, feels Klopp
Klopp praised his team for staying "concentrated" - which he lauded as "very important" and "maybe the most important thing" - as ensuring they didn't slip up further than a draw.
And he also took heart from the fact that "everybody could see" so far this season that Liverpool are "able to score fantastic goals" on "a very good day."
But the boss admitted that his side found it "much more difficult" against Southampton due to "different reasons", insisting that "to stay in the game is the best news we can get" and that while he's not "happy" with the result, he is "fine" with it.
Klopp insisted that his team were up against a "good team" and suggested it was one of the "more difficult games" for Claude Puel's side to play this term.
He revealed that he told his team at half-time that they had "already played more the game we wanted to play than they [Southampton] could" - insisting Puel's team "usually" adopt a different style which sees them "have the ball much more" and "leave two or three players up for counter-attacks."
"It was not often [they did that] today because we did what we did," Klopp continued, saying that his team were "patient enough" but that they were "not clear enough" with their ideas in the first period.
Yet he said that they "did better" after the break and that "when you play out a draw, it's better that you should have won than should have lost because of the performance", again saying he is "fine.
We performed better than we expected, adds Reds boss
On the Reds' attacking approach in the closing stages, Klopp added: "Obviously we wanted to win. I thought we protected our offensive movements pretty well. Joel [Matip] and Dejan [Lovren] did really well in these situations."
He felt that Liverpool "took a little bit more risk" late on after their substitutions, insisting the game "felt a little bit like it's a final" because it was "intense for all of us."
Klopp noted the various instances in which his side "could have scored" - highlighting the chances that Clyne, Firmino, Coutinho and Mane all had - but said it was "really good" that they "had these moments."
He reiterated that "the result and the performance" are the "two things" that are "very important" in football, also adding: "The result is not perfect, but the performance was much better actually than we could have expected, when we think about our preparation for the game and when Phil and Roberto came back."
Klopp admitted that it is "nearly the same" with all the teams with players on international duty over the two-week break, however calling it a "little bit more difficult" with players from South America because "they have long flights and all that stuff."
Expanding on his point, the Reds boss stated that Brazil duo Coutinho and Firmino were back at "the earliest moment" they could be on Thursday but could not train after a long flight.
He said that Mane "came back pretty early" and was "all good" because he was back on Merseyside on Monday, but said that Coutinho "had a little bit [of a problem]" due to the flight - a problem he dismissed as "obviously no big problem" because he started.
Klopp added: "We would have taken no risk. 100 per-cent not. Three things are important for me - what the doctor says, what the scan says when you have one and what the player says, and all were thumbs up so we let him run."