Besic's deal was confirmed yesterday.
Martinez told evertontv: "We have followed him for a while. He was playing for Bosnia Under-21s the first time we saw him.
"He played with an impressive arrogance and an impressive understanding of how to play from the back and having real strong concepts in his positional play and technical ability.
"The way he played against top players in the World Cup - the way he marked Lionel Messi, the way he played against Argentina, the way he played in a really open game against Nigeria - they were really good games to evaluate the value of 'Mo' and the potential he has as a player at such a young age.
"He fits perfectly well into the type of player we need and want at Everton."
Besic shone in Brazil as a defensive midfielder but also has plenty of experience as a central defender with his former club side Ferencvaros.
"Where we play him will depend on our needs," said Martinez. "What we are trying to do is get a really strong squad where the competition levels are huge and now we are getting that.
"We have been seeing a lot of good partnerships all over the park and Mo Besic will fit in extremely well and will add a needed experience, especially in the Europa League."
For the big stories and the banter follow us on Twitter: @tribalfootball