The Reds announced Karius’ departure minutes after the final whistle in Saturday’s 1-0 win at home to Brighton, with the 25-year-old arriving in Turkey earlier in the day.
Karius will spend the next two campaigns in Istanbul, and will join Besiktas on a permanent basis if he meets certain performance-based criteria.
This comes after Klopp’s decision to replace his troubled compatriot with Alisson, who moved to Merseyside in a then world-record £65 million deal with Roma in July.
The manager has insisted plans to sign the Brazil came prior to Karius’ nightmare performance in the Champions League final.
But nonetheless, speaking at his post-match press conference at Anfield, Klopp acknowledged the reprieve Karius would enjoy at Besiktas:
“We said goodbye yesterday, he’s 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.
“A lot of people spoke now about our improved defence, and as good as Alisson was in the last three games, in most of the games when we were already really good he was not here.
“It was Loris in goal, and he did a lot of good things.
“When I saw the Besiktas team, it’s quite an experienced team so they have a good chance to win the league in Turkey.
“Now they have a really good goalkeeper, I’m really happy for him and he was very happy for the opportunity as well.”
Fulham and Napoli both baulked at the finances involved in signing the Belgian, and now Klopp is planning for the season with him as No. 2.
“It’s all good. We have Simon Mignolet here: a wonderful, perfect goalkeeper as well for us,” he added.
“So that’s the situation now, plus the two kids of course.”
Alisson has now kept three clean sheets in his first three competitive games, with his performances further vindicating the big-money fee paid.