Karius, 25, posted a heartwarming message on his social media challenge after being unveiled as the Turkish club's latest signing.
He said on Instragram: "A big 'Thank You' goes out to all of the Liverpool fans for the support until now – you indeed never let me walk alone."
The German stopper endured a nightmare in the Champions League final for the Reds, twice making calamitous errors leading to Real Madrid goals.
He was ridiculed on social media and sent death threats as his mistakes cost Liverpool a sixth European Cup success.
Karius then made another error in pre-season before Jurgen Klopp splashed out £67million to sign Roma and Brazil goalkeeper Alisson.
The German joined the Greek giants on a two-year loan deal from the Anfield club, which was delayed by a four-day public holiday in Turkey.
He was given a hero's welcome when he landed in Istanbul, though, as he looks to resurrect his career after that night in Kiev.
In his social media posts, the former Mainz player added: "I'm proud to be part of the famous @besiktas family now.
"Your welcome at the airport yesterday was simply overwhelming! I will give it my all to win as many titles as possible with Besiktas."
And Klopp passed on his best wishes to the outgoing Karius following the Reds' 1-0 win over Brighton.
Klopp said: "He is really looking forward to it, so that’s good. It’s good for him, I really think. Loris is a fantastic goalkeeper at a very good goalkeeper age.
"When I saw the Besiktas team, it’s quite an experienced team they have there. They have a good chance to win the league in Turkey, that’s cool.
"Now they have a really good goalkeeper. I’m really happy for him and he was very happy about the opportunity as well. All good.
With Karius out on loan and Danny Ward joining Leicester this summer, Simon Mignolet is the only experienced understudy to Alisson, but he too has expressed his desire to leave the club.
Liverpool sit top of the Premier League table with three wins from three matches, ahead of Chelsea and Watford on goal difference.
And to bolster their chances of securing their first league title since 1990, they have recruited world-record holder and specialist throw-in coach Thomas Gronnemark to help out the first team.
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