The German was Jurgen Klopp's first-choice keeper at the end of last season but made two costly errors in the loss to Real Madrid in Kiev.
Liverpool signed Alisson from Roma in the transfer window to take over from Karius, who has now been given a chance to start afresh in the Turkish Super Lig.
Karius, who tests later revealed had suffered a concussion during the final against Madrid, may be forever be associated with his high-profile gaffes, yet the 25-year-old wishes to consign them to history.
"This is life, this is football," Karius said.
"If you give the circumstances - with the serious injury as well - that you don't know in what way it affected me, you have to move on.
"It was an unlucky day, but it's football. I don't think a lot of players reach a Champions League final. If you look at how many goalkeepers in the league reach a Champions League final, I don't know. So, it was a great achievement.
"Unfortunately, on this day we were unlucky and it wasn't the best performance. But there was a lot of things that came together, it's in the past now and I look forward to the future."
Karius also revealed Klopp wanted him to remain at Anfield but that a move away from Liverpool was "important" for his career.
"He told me he would like to have me in the squad because he wants to have the best quality players," Karius added.
"I asked him for my wish and that is important for me to make this move and play for Besiktas. I talked with him a long time. He understands my situation.
"I think it was a good deal for everyone. He wished me well but he said he would have liked to keep me."