The 25-year-old was named on the bench for Liverpool’s opening Premier League fixture of the season against West Ham, but left out of the squad against Crystal Palace and Brighton.
On Friday, Besiktas appeared to confirm the arrival of the Liverpool keeper with a graphic of him on their Twitter page, and the move was completed on Saturday.
The Turkish club have started their Super Lig campaign with two wins and are tied 1-1 with Partizan in the Europa League play-offs.
Karius made 49 appearances for the Reds after joining from Mainz in 2016.
Thirty-three of those appearances came last season, where he came under fire for his performance in the Champions League final, which included two errors that led to goals in the 3-1 defeat against Real Madrid.
He was in tears at the final whistle and the Reds decided to sign Roma’s Alisson, subsequently signing the Brazil ‘keeper in a then world-record £67m transfer.