Goal revealed earlier on Saturday that the 25-year-old was heading to Turkey to complete a move away from Anfield.
And Karius has now been allowed to leave and rebuild his career in the Turkish Super Lig side after agreeing a two-year loan to Besiktas.
Karius was named on the bench for Liverpool's Premier League opener, a 4-0 defeat of West Ham, but was omitted from the squad for Monday's trip to Crystal Palace and Saturday's win against Brighton and Hove Albion.
He made 49 appearances for Liverpool after arriving from Mainz in 2016.
"Everybody at LFC wishes Loris the best of luck during his loan spell," read a Liverpool statement.