Besiktas said in a separate statement that Rangers had paid four million euros ($4.09 million) for the 21-year-old.
"I'm delighted to be joining a club the size of Rangers and am really looking forward to meeting my new teammates," said Yilmaz, who has won the Turkish league with Besiktas and has Champions League experience.
"This is a club who have shown they can compete against some of the best teams in Europe and I'm excited by the challenges we have ahead of us."
Manager Giovanni van Bronckhorst continued: “I am really pleased to have added Ridvan to our squad and I believe he will bring a lot of quality to the club.
"I have been really impressed with his development in a really demanding, winning environment at Besiktas over the last few seasons.
Yilmaz made his Turkey debut last year and has six caps.
Rangers, Scottish champions in 2020-21, finished second behind Celtic last season and lost the Europa League final to Eintracht Frankfurt.
Giovanni van Bronckhorst's side begin the 2022-23 league campaign against Livingston on July 30.