On the night when the Reds were securing a 5-1 aggregate win over Manchester City, the Serie A side produced a major shock in overturning a 4-1 first-leg deficit to beat Barcelona on away goals.
With the last-four line-up completed by 12-time winners and defending champions Real Madrid and Bayern Munich, who have lifted the trophy five times, real pedigree potentially awaits Klopp’s team.
Real have won three of the last four finals while Bayern’s last win was in 2013 - against Klopp’s Borussia Dortmund - after finishing runner-up twice in three years.
By contrast Liverpool, for whom the last of their five triumphs came in Istanbul in 2005, are in their first Champions League semi-final for a decade.
Although Roma are in the last-four for only the second time after finishing runners-up in 1984, Klopp was impressed with the manner of their comeback against Barcelona.
Drawing them would produce an early reunion for 39-goal Mohamed Salah, who left the Italian capital in the summer in a £38 million deal which is now looking good value.
“I walked up the stairs and somebody told me (Roma had gone through) — I really thought it was a joke,” he said.
“Not that I don’t respect Roma, the absolute opposite. They have a fantastic team. They lost Mo Salah and they are in the semis, that’s quite a big thing.”
Of the clubs that remain in the competition, Liverpool have the best record against Real, for whom Cristiano Ronaldo scored his 120th Champions League goal with an added time penalty to end Juventus’ hopes on Wednesday.
Despite two defeats in their last two Champions League encounters in 2014 — one of which saw then manager Brendan Rodgers play a weakened side in the Bernabeu — in their three meetings prior to that the Reds have three wins, including the 1981 European Cup final in Paris and a 4-0 humbling at Anfield inspired by Fernando Torres and Steven Gerrard.
Their seven previous encounters have been less clear-cut with two wins and four draws, with their last meeting a 3-2 victory in the 2001 European Super Cup after Liverpool won the Uefa Cup.
Klopp, of course, has more recent history with Bayern as Dortmund manager having lost the 2013 final to Arjen Robben’s 89th-minute winner.
Liverpool’s last of five meetings with Roma came in 2002, with a 2-0 win putting them into the quarter-finals of the Champions League.
Klopp insists they are focusing on themselves rather than their opponents as the prospect of a first European Cup final since 2007 - having reached the Europa League final two years ago - becomes a realistic prospect.
“I don’t care who is in (the competition). We are here and we deserved it, and I am really happy about that,” he said.
“So far, we fought for each opportunity to get it - and then to use it is so hard. But that’s what we will do.
“It’s obviously a while ago that Liverpool were in the semis. It was a while ago that I was in the semis (in 2013).
“Now we are there together — so that’s quite a cool moment and I’m really happy about it.
“The other teams will have their targets as well when we meet them in the semis; I think they are not without ambitions.”