The Brazilian midfielder's transfer to Al-Ittihad on Monday brought an end to his hugely successful five-year spell with the Reds.
Referred to as the team's 'lighthouse' in the centre of the pitch, Fabinho was a key and dependable figure who made 219 appearances and lifted a host of football's biggest prizes, including the Champions League and Premier League.
Paying tribute to the 29-year-old, Klopp told Liverpoolfc.com in Singapore: "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.
"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."