The five-time European champions will take on Ajax or Tottenham in Madrid on June 1 after digging deep to overturn last week's 3-0 first-leg defeat at the Nou Camp.
Divock Origi put Liverpool ahead after just seven minutes and the hosts kicked on in the second half as Anfield witnessed a night that will live long in the memory.
Georginio Wijnaldum came off the bench to score a quickfire brace, before Origi turned home Trent Alexander-Arnold's smart corner to seal a 4-0 win and progress 4-3 on aggregate.
WATCH: RTÉ Soccer panel react to amazing Liverpool win
Didi Hamann, Liam Brady and Damien Duff react to Liverpool's famous 4-0 win over Barcelona, which sets up a Champions League final against Ajax or Spurs.
Asked where this remarkable victory ranks, Liverpool manager Klopp said with a smile: "Top three. "It's a special night, very special. Winning against Barcelona is obviously one of the most difficult things in the world of football.
"Winning against Barcelona when you're 3-0 down, makes it not easier. We had to score four goals and was not allowed to concede pretty much.
"We made it more difficult, so we didn't really think about it to be 100 per cent honest. We tried to build on the performance in Barcelona and all the good things we did there and try to win the game step by step.
"We attacked with whatever we have and can throw on the pitch.
"That made it a really special game, it made us really difficult to play against tonight. What the boys did and that was this mix of again a big heart and football skills. It's unbelievable.
"But then we know, and we didn't learn in the first game and we knew it before already, if you have chances, you have to score because otherwise you get punished.
"Tonight we scored, in different ways. I said it so often now but the thing that made it really possible – and I said to the boys before 'I don't think it's possible but because it's you I think we have a chance' – they are really mentality giants. It's unbelievable.
"After the season we played, the games we had, the injuries we had now in this moment, if you go out there and ask who bet a penny on us, I don't think you find a lot of people.
"So, and then going out there and putting a performance like this on the pitch is unbelievable. I am really proud to be manager of this team. It's unbelievable what they did tonight. It's so special.
"I will remember it forever 100 per cent because I don't know if it happened before or it can happen again. I really don't know. The boys did it, so it was brilliant."
Liverpool's spectacular victory has them dreaming of two trophies.
While Manchester City are in the driving seat for the Premier League title heading into Sunday's finale, the Reds will look to capitalise on any slip up by beating Wolves.
Klopp said his players did not want to talk about City's win against Leicester ahead of Tuesday's match at Anfield, where they sealed the chance to make amends for last season's Champions League final loss to Real Madrid.
"The plan was to go to Madrid and get it in there finally," Klopp said.
"That was the plan actually and now we go there in three weeks' time. I am really looking forward to that. With the Premier League, nothing changed really. We knew we have to win the game against Wolves and then we will see what happens around."