The German's side travelled down to the Amex Stadium seeking a slice of redemption for last weekend's excruciating 2-1 loss to Tottenham Hotspur, which ended their unbeaten streak for the 2023-24 season.
However, Liverpool were off colour from the first minute and gifted Brighton the opener on a silver platter, as Simon Adingra calmly finished from range after intercepting Virgil van Dijk's attempted pass to Alexis Mac Allister.
The Reds unexpectedly turned the game on its head just before the break, though, as Mohamed Salah curled home the equaliser before finding the bottom corner from the penalty spot in added time.
This time, it was Brighton's turn to rue a defensive mistake, as Bart Verbruggen - whose decision-making throughout was questionable - played a short pass to Pascal Gross when the German was under pressure from Dominik Szoboszlai.
The German had no choice but to bring Szoboszlai down by pulling his shirt from behind, but the referee did not blow for a penalty until Carlos Baleba had also cleaned out Luis Diaz a couple of seconds later.
Gross's challenge led to a debate over whether the Brighton man had denied Liverpool a clear goalscoring opportunity, which would have constituted a red card, but the 32-year-old escaped without so much as a caution.
Substitute Ryan Gravenberch then squandered a wonderful opportunity to put his side 3-1 up early in the second half, clipping the post with the goal at his mercy, before Lewis Dunk caught Liverpool napping from a free kick and stole a point for the Seagulls.
When asked whether he felt Gross deserved to be sent off, Klopp admitted that he did not consider the denial of a goalscoring opportunity rule at the time, but the 56-year-old jokingly claimed that he is "too old" to lose sleep over such things.
"I didn't realise it, to be honest. I saw it was a penalty and didn't think about red, but I heard now it is about goalscoring opportunity," Klopp said in his post-game press conference.
"If it is a goalscoring opportunity, we can ask the question here: what do you think, was it a goalscoring opportunity? Hands up. I didn't see it back... we have a few hands up, by the way! But what can I say about that? I am over it, I am too old for these kind of things. We will not change it anymore."
When questioned on whether he thought a point apiece was a fair result, Klopp added: "I think unfortunately, yes. I think it is clear in the second half we should have scored for 3-1, there were one or two really good opportunities but because we don't score there, we keep the game open and in this area where the free-kick happened we produced too many set-pieces, corners and especially free-kicks.
"I saw it back and if one of our boys touched the ball I think it was an own goal. That's what a set-piece it was. Then Dunk was there and could score from there. It was intense for both teams, so I think it's the right result in the end.
"I know some moments they had the upper hand, but that's it and I really think if we had scored the third one, we would have deserved to win; definitely. But I guess it was like it was."
Despite dropping another two points on the South Coast, Liverpool are in a respectable fourth place in the table heading into the international break and return to action with a Merseyside derby against Everton at Anfield on October 21.