A strike inside the opening three minutes from Richarlison put the Blues in front before the Reds lost Jordan Henderson to injury midway through the first half.
Victory was secured for the visitors in the closing stages, with Gylfi Sigurdsson netting from the penalty spot.
Afterwards, Klopp addressed the media in his post-match press conference. Read on for a summary of what he had to say…
On how much the result hurts...
A lot, a lot. But, we conceded a completely unnecessary first goal and we should not forget that. That's a big part of the game. There were two big parts. You have to defend and we have to score. In one situation we didn't defend well enough, so they could score, we made a mistake if you want. And we didn't use their mistakes or we didn't use the things we created. And that's why we have the result.
On the decision to award Everton a penalty...
I really think it's unfair to ask me because everybody asked me that question. That means everybody thinks it was not a penalty. But that's not really important because the ref thought it's a penalty. I wanted to talk to him now after the first few interviews I had but they left already. I just wanted to ask what did he see because the VAR calls you over in a situation like that, then I think he is in doubt about the decision. But he needed only a second. He went there, watched it from three or four yards, turned and penalty. So he obviously saw something that all other people didn't see. I didn't see it back yet but everybody who speaks to me tells me the same, 'How can it be a penalty?'
On whether he can rationalise losing four games in a row at home or whether it's too big a picture to do so...
Not all the games were the same obviously. [We were] dominant in most of the games if I am right, we lost an away game as well where we were really dominant against Leicester. That means in the decisive moments we make either a mistake or just something strange happens, so that's how it is. If we make a mistake, it's 100 per cent they score from it and if we cause them to make mistakes that doesn't mean immediately we will score. I think everybody who saw the game tonight would say, 'OK, if we watch the game without the goals, how can that happen?' But it happened and not the first time, so we are really critical with ourselves there. How I said, the first goal was our mistake and we don't finish the situations off. It's about us as well, it's not who can we criticise for that, that's clear, we know that. I said it already in other interviews, I don't like to talk tonight about the good parts of the game because we lost the game and we lost the derby. But from tomorrow on we have to use that again, that's how it is, because we cannot change the situation by playing bad, that's not possible. But in decisive moments we have to improve, that's clear.
On how the players are handling this period...
We had some problems, you know them, and again if you saw an attitude problem today on the pitch, for example, then you have to write it. If you didn't see that then you cannot write it. We have to play but we have to force it as well, that's how it is. In the decisive moments, we have to be calmer, that's how it is – like in the finishing situations and I don't mean now the headers from Sadio, which were really tricky with circumstances we had tonight, or the chance from Mo where it was super play with Shaq and Trent when Mo is alone and that's then really difficult because Pickford is in front of him, short distance. But we had other situations where we had really finishing opportunities and we didn't finish the situations off. So, the only way I know is to try it again and again and again, and I don't see any relation to the last three years. That we were champions last year, that was already a month after we were champions, it didn't feel like it was a month ago – that has nothing to do with that. The boys are still full of desire, I see that. But to change a football game, a result, to get the result you have to be decisive in the right moments defensively and offensively – and that's what we are lacking.
On Liverpool's defensive options and how difficult the recurring injuries are…
We have options, of course, but how difficult it makes it… it makes it really difficult. But, for example, Nat was for sure not for one second our problem and even Hendo in the beginning of the game was really dominant, especially offensively. So, yes, it is a massive blow losing Hendo again. Massive. But Nat played a super game. Again, we meanwhile unfortunately got used to it that we have to change things and we just try to set up again for the next game. As long as we have 11 players we will do that.