The 21-year-old, who featured for Bosnia at the World Cup, linked up with his new squad for their summer trip to Thailand last week and has now completed the move which Goal understands is worth £4 million.
Besic watched from the sidelines as Everton lost to newly promoted side Leicester City in Bangkok at the weekend and Martinez is delighted to be able to finalise the deal.
"Muhamed is a young footballer who has developed massively in the last two seasons. He has been playing as a centre-half and as a defensive midfielder at an incredible level," Martinez told the club's official website.
"What he did with Bosnia and Herzegovina in the World Cup, he showed that he is a really mature footballer, that he covers the ground really well, that he is very dynamic and technically very gifted.
"He is bringing us really good strength in an important part of the side."
Besic becomes Everton's second summer signing after fellow defensive-minded midfielder Gareth Barry penned a permanent contract earlier this month.