30-year-old Fabri worked with Fulham goalkeeping coach Jose Sambade Carreira at Deportivo La Coruna.
Last week saw Fulham's back-up goalkeeper David Button join Brighton, leaving just Marcus Battinelli, who was the first-choice keeper for the second half of the season.
Fabri told the BBC: "I'm happy to have the opportunity to be in the Premier League."
The goalkeeper helped Besiktas win the Turkish Super Lig title in 2017, conceeding just 30 goals in 34 league matches last season as the side finished fourth.
He also helped Besiktas reach the last 16, bringing his number of Champions League performances up to 11.
Fulham will have the option to extend Fabri's new contact for 12 months, if they click with the goalkeeper.