The 29-year-old has ended his glittering five-season stay at Anfield to join Nuno Espirito Santo's side in a £40m deal, becoming the latest high-profile name to be seduced by Saudi Pro League riches.
During his time on Merseyside, Fabinho was regarded as one of Europe's best defensive midfielders, having played a key role in helping Liverpool collect their sixth Champions League crown in his debut campaign.
The former Monaco man subsequently won the Premier League, Club World Cup, UEFA Super Cup, FA Cup, EFL Cup and Community Shield at Liverpool, as well as registering 11 goals and 10 assists in 219 games.
Fabinho struggled to show his best form in the 2022-23 campaign, but Klopp still paid tribute to the Brazilian's "massive" contributions while emphasising the need to adapt to change.
"Of course we wish him all the best. Fab was for so long the insurance we had more or less in midfield. He always gave us the freedom to play all the fancy stuff," Klopp told liverpoolfc.com.
"He was involved in that from time to time as well but loved to be the hoover for the team, loved to do all the dirty work for the team.
"Together with the centre-halves, when both full-backs were flying right and left, all the strikers, all the midfielders somewhere and then just Fab and the two centre-halves when it looked, 'Oh OK, if we lose the ball, there's still a few players at least who can sort that.' And he was massive.
"So yeah, we will miss him, definitely. But again, it's like it is. Life is about changes - that's always - and you obviously have to adapt to them. If you don't do that, if you stick in the past, something will hold you back - and I'm not that person.
"It was always clear that in these, from my point of view, nearly eight years that things will change. A lot of things [that] changed we met here, Lucas Leiva and Martin Skrtel, and the only player they played together with when they were [here] - at least Martin told me that - is Joe Gomez. All the other players are new.
"So that means in these times a lot of things happened and they will happen. That's all about it. And yes, we will miss Fab - but Fab will be fine, we will be fine and we always will be thankful to each other. That's, for me, very important in life. So all the best, my friend, and hope to see you soon."
Fabinho has jetted off to the Middle East not long after Jordan Henderson relinquished his duties as Liverpool captain to join Steven Gerrard's Al Ettifaq on a reported £700,000-a-week contract.
Liverpool have also lost Leighton Clarkson, Naby Keita, Alex Oxlade-Chamberlain, James Milner and Juventus loanee Arthur from their midfield ranks this summer, while Thiago Alcantara is now in the final year of his contract.
After ending their interest in Real Madrid's Jude Bellingham, Liverpool spent £95m on the double signing of Alexis Mac Allister and Dominik Szoboszlai and are expected to table an improved bid for Southampton's Romeo Lavia, having already failed with a £37m approach.
The Reds - who named Virgil van Dijk as their new captain and Trent Alexander-Arnold as their new vice-captain on Monday - will play their penultimate pre-season match against Bayern Munich in Singapore on Wednesday.