Everton manager Roberto Martinez confirmed last week that his club were on the verge of finalising a deal to sign the 21-year-old, who linked up with his new team-mates on their pre-season trip to Thailand.
And Besic's arrival at Goodison Park was officially announced on Monday, with Ferencvaros having received an undisclosed fee for a player who featured in all three of Bosnia's matches at the recent World Cup.
A delighted Martinez told Everton's official website: "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.
"What he did with Bosnia-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 became the youngest player to be capped by Bosnia at senior level when he made his international debut against Slovakia in November 2010 at the age of 18.
After starting his career with Hamburg, he joined Ferencvaros two years ago and went on to make 47 league appearances for the Hungarian side.