The future of the German shot-stopper had been the subject of intense debate following his high-profile errors in last season's Champions League final, while further mistakes in pre-season only fuelled speculation.
Alisson Becker's arrival at Anfield saw Karius fall down the pecking order, and the 25-year-old completed his switch to Besiktas shortly after Liverpool's 1-0 win over Brighton & Hove Albion this afternoon.
"We said goodbye yesterday, He is really looking forward to it, it is good for him and I really think Loris is a fantastic goalkeeper at a very good age," Klopp told reporters at his post-match press conference.
"A lot of people spoke about our improved defensive things and as good as Alisson was in the last three games we were good when he was not here and Loris did a lot of good things.
"Besiktas have a good team who have a good chance to win the league and they now have a really good goalkeeper and I am really happy for him."
Karius will initially join Besiktas on a two-year loan deal for a £2.25m fee, with the Turkish club obligated to make the move permanent for a further £7.25m in 2020 if certain criteria such as appearance targets and European qualification are met.