Leroy Fer's finish from a corner-kick gave the Swans a deserved lead at half-time with Francesco Guidolin's men having been much the better team.
But a stirring response after the break saw Roberto Firmino restore parity in the 54th minute, flicking a header into the far bottom corner from Jordan Henderson's lofted cross into the box.
After Firmino was felled inside the area late on, James Milner stepped up to dispatch his third penalty in just two games and hand the Reds a fourth straight victory in the Premier League.
Klopp says his side's first-half display made "no sense"
On the first-half performance, Klopp insisted that his team "were not ready" as evidenced by their "bad" display in the first 45 minutes.
He acknowledged it as his "responsibility" for the team to be "ready for games" but added: "Today I saw we weren't. [The players'] body language was not good. The build-up was absolutely too static."
The Reds boss said that Swansea also "did well" - which he said was perhaps "the most important thing" of the first period - but bemoaned how his side were "really too static" and how "easy" it was for the Swans to "disturb" their build-up play.
Klopp declared that Liverpool were aware of what the home side "were going to do" and plan against it in training, calling it "pretty clear" that they "would be similar to what they did against [Manchester] City."
"We knew where the spaces were but obviously it was a kind of misunderstanding," he continued, insisting that they "didn't do well" and "were never compact enough for counter-pressing situations."
Klopp also criticised his Liverpool team's play for making "absolutely no sense" because they "lost a lot of balls" and 'were not really in the [right] situation" to respond and retrieve possession.
But he felt that while his side weren't "immediately better" after conceding the opening goal, they were "better" as the first-half wore on - explaining: "We could see that when we were in the game, we let the ball roll and our movements were better, then it was difficult for Swansea."
Klopp admitted that they didn't create "the biggest chances" but had "the moments" that gave them "important information for the second-half" which he felt was "better" even though they "were still not brilliant."
The German also felt that Liverpool "deserved the win" even in spite of the late chance that Mike van der Hoorn spurned, despite insisting that had the defender scored then it would have been their own fault and Swansea would have "deserved the point for sure."
However, reflecting on the second-half as a whole, Klopp said that they "were the better team" overall and called it a "deserved" three points.
Reds boss: We were angry with ourselves at half-time
Asked about how angry he was at half-time, as a result of the below-par showing from his side, Klopp revealed that he told his players the scoreline made "no sense" because they "could have done so much better in the first-half" and "could still do better in the second-half."
While he acknowledged that losses will happen, he insisted Liverpool cannot "accept a bad day" and must still "fight against the [bad] start" and "have to strike back in the game and cool down."
Klopp mentionted that there were "small things" where it was evident his side had "lost patience with ourselves", such as when they were "moaning about the decisions when the referee was absolutely right."
He said that the team needed "to stop this" and suggested that half-time came at the right time, because they were "angry" but "with ourselves, nobody else."
"We had to show we can do better because [at that point] nobody could see why we were here," insisting that they looked like they were only there to turn up rather then "to win a football game" and said that there "was no doubt about this anymore" in the second-half.
Reds are still learning, admits Klopp after poor start
He told reporters that he usually "cannot change a lot in a game" but said at half-time on this occasion he could "change a lot" particularly because his players "have open ears."
Klopp declared it "very important" on days when they fail to produce a "brilliant performance" that they are "still able to win" as he went on to hint that they will face more games like this throughout the season.
"It was very important that we showed the reaction in the second-half," he continued, insisting that training in the week gave no indication that "this would be the day where we would be that bad."
Yet Klopp acknowledged it showed his side's need to keep "learning" and said that they "have enough time to think about it now until we play again" as he accepted his "responsibility" for "the start" which he was "not happy with."