The 21-year-old travelled to Thailand with the Toffees for their pre-season match against Leicester at the weekend after an agreement was reached with the Hungarian side.
Besic officially completed his move after the finer details of the paperwork were concluded.
"Muhamed is a young footballer who has developed massively in the last two seasons," manager Roberto Martinez told evertontv. "He has been playing as a centre-half and as a defensive midfielder at an incredible level.
"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, who became his country's youngest senior international with a debut aged 18, played three times at the FIFA World Cup™, with one of those appearances coming against eventual finalists Argentina.
Born in Germany, he began his professional career at Hamburg before transferring to Ferencvaros two years ago.