Beaten 3-0 in Barcelona on 1 May, Divock Origi and Georginio Wijnaldum both bagged a brace as the Merseysiders routed Barcelona 4-0 at Anfield.
"With any other team I wouldn't think it was possible," said Klopp. "The players are really mentality giants. Going out there and putting in a performance like this on the pitch is unbelievable. I am really proud to be the manager of this team.
"What they did was really special. I will remember it forever."
Liverpool can clinch the Premier League title on Sunday if they beat Wolverhampton Wanderers and Manchester City fail to win at Brighton.
While that trophy depends on the skills of Pep Guardiola's City players, Liverpool will play either Ajax or fellow English Premier League outfit Tottenham Hotspur at the Wanda Metropolitano.
The Dutch side take a 1-0 advantage into the second leg of the semi-final in Amsterdam on Wednesday night.
That will seem slender after Liverpool's feats in front a fervent crowd. Origi scored his first after seven minutes. But Barcelona survived the early barrage to go in at half-time 3-1 up on aggregate.
Wijnaldum, on for the second-half for the injured Andrew Robertson, changed the complexion of the tie with two goals in two minutes to bring parity on aggregate.
Barcelona never recovered and Liverpool's winner in the 79th minute highlighted their mental disintegration.
There were nine Barcelona players in their penalty area as Trent Alexander-Arnold prepared to take a corner. But none of them had picked up Origi. Alexander-Arnold's quick drive into the box was turned into the net past the startled Barcelona goalkeeper Marc Ter Stegen.
"A genius moment," said Klopp of Alexander-Arnold's improvisation. "I saw the ball flying in the net and had no idea who took the corner and who scored the goal.
From then on, Barcelona lacked the collective gumption, guile or the genius of Lionel Messi to redeem their demise.
"We know this club is the mix of atmosphere, emotion, desire and football quality," added Klopp. "This club has a big heart. The heart was pounding like crazy, you could hear it and feel it all over the world. I am so happy we could give the people this experience."
Barcelona boss Ernesto Valverde was left to ponder the second embarrassing European collapse during his time in charge.
The Spanish champions blew a 4-1 first-leg lead to bow out at the quarter-finals against Roma last year.
"The most hurtful thing is to repeat it," he said. "When you have a collapse of this nature we will have a few horrible days ahead."
Barcelona, who have already secured the La Liga title, will have two weeks to lick their wounds before they take on Valencia in the final of the Copa del Rey on 25 May.