The 21-year-old impressed for Bosnia and Herzegovina at the World Cup over the summer and has now sealed a transfer from Ferencvaros.
Martinez's first season in charge of the Toffees last term met plenty of praise and Beric is excited to play for the former Wigan boss.
"The most important reason why I signed for Everton was Roberto Martinez – I think he is a fantastic coach," the youngster told the club's official website.
"I know Everton is a really traditional club and the supporters are fantastic here. I think it is the biggest step for me after leaving Hungary."
Besic's first assignment at the World Cup was keeping a check on three-time Ballon d'Or holder and eventual Golden Ball winner Lionel Messi and the German-born midfielder says that his performance against eventual finalists Argentina brought about much of the interest in his services.
He remarked: "The World Cup was really important and I think the most important game for me was the Argentina game. I think I did a good job and many clubs were looking, including Mr Martinez.
"I think my style is suited to playing in the Premier League and I think this is the best league in the world," Besic added.
"It will be difficult for me because we have fantastic players but I think, with the help of the coach, I will do my best.
"I can play anywhere [in defence or midfield] but I think I will play as a central midfielder for Everton. I enjoy playing in midfield because I get to see a lot of the ball and I think it is my best position."