The 30-year-old has signed a three-year deal at Craven Cottage, where he will be reunited with goalkeeping coach Jose Sambade Carreira, who he worked with during a spell at Deportivo La Coruna.
Fulham have the option to extend Fabri's contract by an additional 12 months and the player is determined to capitalise on the opportunity awaiting him as he aims to break into Slavisa Jokanovic's team.
"I'm delighted to be at an historic club like Fulham, the oldest club in London," Fabri told the club's official website.
"I'm very happy to have the opportunity to be in the Premier League with Fulham.
"Everything is very positive for me and I want to take the opportunity to make the most of this goal of being in the Premier League."
Fabri made 34 appearances for Besiktas in 2017-18 as the club finished fourth in the Turkish Super Lig, a year after winning the title.
He also played in the Champions League and picked up two Turkish Super Cup runners-up medals during his time with Besiktas, who he joined from Deportivo in 2016 after helping the Spanish club to promotion from the Segunda Division in 2013-14.
Khan said: "I'm pleased to announce that Fulham have completed the permanent signing of goalkeeper Fabri from Besiktas.
"He is a tremendous player that we've targeted for some time.
"His scouting profile and his data profile are both strong, and he has exceptional career credentials including Champions League experience, plus he has worked well with our first-team goalkeeping coach Jose Sambade Carreira in the past, so we expect his addition will make us a more complete and competitive football club."