Sadio Mane's 94th-minute header saw the Reds complete a turnaround that maintained their six-point lead at the top of the table.
Trezeguet had given Dean Smith's Villa the lead midway through the first half before Roberto Firmino then had a goal ruled out for offside.
The visitors continued to fight for a way back into the game, though, and drew level in the 87th minute through Andy Robertson before Mane's heroics right at the end.
Read on for a full transcript of Klopp's post-match press conference...
On the emotions that followed Mane's goal…
The best possible in football. Really satisfying. The routine [for Mane's goal], we'd tried before a couple of times and we thought it could be a space where we could make a little advantage. It was not the only routine we had, but it was one I saw in the game two or three times before. We worked yesterday on that and then if you can use it in a very decisive moment then it's really good. Sadio [also] crossed the ball for Robbo's goal. Being there for that goal after the situations he had in the game when he got a yellow card – I think when you see it back, it is for sure not diving, there is contact, but it is the decision you have to accept. In the moment, then the crowd takes you a little and try to make you even more insecure – but that's obviously not possible with Sadio. He stayed in the game, got the yellow card there, but still stayed in the game even defensively. Very good.
On whether it was Liverpool's 'most difficult game' of the season so far…
We made it difficult for ourselves. We started playing football good but not exactly like we should have done. Aston Villa were ready today for a proper fight, a proper battle, defend with all they have, try to find spaces for counter-attacks and be there and have good set-pieces. We played good football but didn't finish the situations and then we conceded a goal, then it is not so easy to change these wrong decisions, this wrong path, immediately. We needed a bit of time, but we had good chances as well. I don't think they had too many, but we had good chances. Second half then we started immediately much more aggressive in the right manner, our counter-press was better and stuff like that. Then we could change twice, fresh legs and shooting from distance – Ox helps massively with situations like that. We brought Sadio to the right side, Ox left in the half-space, opening the space for Robbo then going inside. That was really good. I think, of course, it is lucky when you score these goals but I think if you watch the game, then I think if one team deserved to win it then it is us. But I can imagine how bad Aston Villa feel because with all they did today they would have deserved something, but that's how it is sometimes.
On whether he felt the game was getting away from Liverpool at any point…
I don't have time to think about things like this, to be honest. I am not there to feel happy and disappointed and sad and angry or whatever. Yes, sometimes I am angry, but not because I plan it. Sometimes I am disappointed, but not because I plan it. I never thought a second about 'did we lose already?' or 'can we win it still?' It is not like this. We just have to try to score, that's everything – and that's what obviously we tried. It's not that I think now that we have had a couple of comebacks then today there were would be another comeback – how can you think that? But we know it is possible, everybody knows it is possible. We really know it, the boys tried it. I think we got the free-kick with Trent, then somebody in the wall [deflected it with a] header and then that was the corner when we scored. That's football, you need to have these situations.
On his players having clarity of mind in high-pressure situations...
We don't have any alternatives, what can we do? Do we say, 'OK, today it's too hard'? It would've been a tough game anyway because of the quality of Aston Villa but we helped a little bit. It looked good but I didn't like it. Yes, it's good football but in the end you have to finish it off – and we didn't finish it off. We had one counter-pressing situation after 10 minutes, we didn't win the ball back. But it's all good, these things can happen and it's not a massive mistake, it's just the wrong path. When you concede the goal and now it's getting serious and now we have to be here. We still have to learn, still have to improve – and we do and we will, I know that. Sometimes you need to get knocks, and we got knocks a lot already. For little mistakes we got really big knocks. But in the end, in the moments so far we could sort it most of the time and it's OK. Yes, of course when you score these kind of goals it's lucky, but it's not that we didn't deserve to win the game. I think everyone who saw it thought, 'They are not bad, they can win football games'. I am fine with our overall performance but we had moments which I didn't like.
On VAR ruling out a Firmino goal because his armpit was adjudged to have been offside...
Look, we have now a real problem. We won 2-1, that's why, for me, it's not that serious. But now we talk about serious moments, very important moments in football. It is not the right way that we sit here, talk about it and everybody wants to laugh about. It's not to laugh about, to be honest. For that it is too serious. Managers get the sack for losing football games. It's just like this. I don't want to make it bigger really than it is, but in fact we just have to clarify it. I don't say anybody is doing it on purpose or whatever, we just have to clarify it. My analysts showed me after the game, I didn't see anything and now I heard it. Whoever did it is not doing it on purpose but at the end, we need to sort a couple of things obviously. The handball situation with Targett, nobody talks about that because it was not whistled. Imagine if it would have been whistled and then we would have talked about it. But nobody whistled it, we didn't say anything. Was it a penalty? The shot is here, the shot hits the arm, so in a good old time it was a penalty! Sometimes it's in your favour and sometimes it's against you, that's how it is. That's not a problem. But we have to make sure the new system helps the game and not confuse it, that's all. But I think that's in the interests of all of us.
On the body language his players showed towards the end...
It changed everything. That's a good point. I didn't like the body language in the first half, that's exactly how it is. We were not like warriors, we were like players. That's OK, we have to play. But at the end, you have to win the ball back. That's what I said, with 10 minutes, of course we lost balls, had one counter-press which the impulse was right, three players there but they came out. It can happen because you cannot be 100 per cent successful with counter-press. On the other side, we were really being here, being there but don't finish the situation off. I don't know how long it took until we shot the first time from distance, maybe it was Ox, I am not sure – that would then mean [after] 60 minutes, that's quite a time. Body language is massively important.