Klopp’s side return to European action on Tuesday night for the first time since eliminating Manchester City - considered by many to be the best team in Europe - earlier this month.
Roma pulled off a shock of their own in the quarter-finals however, famously overturning a 4-1 first-leg deficit to knock out five-time winners Barcelona.
Though both were unfancied when the draw for the last eight was made, one of these two teams will contest next month’s final in Kiev and either could add a new chapter to their storied histories.
Liverpool are the favourites but Klopp insisted at his pre-match press conference on Monday that his team will not allow the weight of expectation to get the better of them.
“There is no pressure,” he said. “We are in the semi-final. We are not the team that has to win. We are the team that can win.
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“We really feel the opportunity. It is a big thing. We came here with not a lot of expectation. We only came here expecting to win the games. People ask me if I feel pressure. No. I only feel opportunity. I am really happy to be here.
“We really think we deserve it. That's really important for me,” he later added. “We didn't come through with a lot of luck, like 'Wow, how did that happen?'
“Porto, at Porto, was a massive statement,” he said of Liverpool’s last-16 win. “Did we expect that before the game? Of course not. Who can expect 5-0 at Porto? They come here and we realise how difficult it is to deal with a 5-0. It doesn't happen too often. Will you play all your first team players or not? It's strange.
“Then you play City and win again 3-0. Everybody tells you that was always enough in the history of football, meaning if you don't go through you are the biggest idiots in the world. But we are still here. That's cool, let's use that fantastic situation.”
Klopp’s side are the highest-scoring team in the Champions League this season and have made a habit of blowing their opponents away in short, explosive spells of scoring.
The Liverpool manager recognises this strength in his attack but believes the greater sense of defensive nous shown in the two legs against City represented ‘a big step’ and was key in progression through to the last four.
“It was so massive when we won the City leg, so massive it felt like: 'Wow!' Not that we are surprised but it was such a big step for us, in the manner that we played the games,” he said.
“We all saw it, in the second half here and the first half there, staying in the game. That was all really brilliant and that's what we have to deliver again.
“That brought us to the semis, not that we can score one, two, three in a row. In the moments when we are not in charge of the game, we are still in the game. That's a very important thing and that helped us massively.”
The build-up to Tuesday’s semi-final has been dominated by memories of the 1984 European Cup final, contested between the two clubs in Rome.
Liverpool, inspired by captain Graeme Souness and Bruce Grobbelaar’s wobbly legs, famously won on penalties after a 1-1 draw at the Stadio Olimpico.
Klopp is aware of this tie’s history but, just as before the two legs against City, he urged his team to put past glories to one side and create a history of their own.
“Most of the players were not born then,” he initially noted. “Was Milly [aka James Milner]? Maybe Milly, I am not sure.” Milner was born in 1986. “Oh, maybe he just looks older.
“Yes, if it helps we will use it but I think the boys need real things, not the good old stories, even though they were brilliant. We only talk about it because we hope it helps.
“‘The spirit of Rome’. In Germany we talk about the spirit of the 1954 and '74 camps [World Cup winners], but it is only to create your own history. That is what we have to do, not what we want to do.”