The 21-year-old, who played in all of Bosnia-Herzegovina's matches at the World Cup, has agreed a five-year deal with the Merseyside outfit. Having joined the club’s pre-season tour in Thailand last week, he was finally unveiled at Everton’s training complex, Finch Park, on Wednesday.
Dzeko, who is the Bosnian national team's record goal-scorer and their third-most capped player of all time, spoke extremely highly of the youngster following his arrival.
"He's a very young player, he will definitely be a big signing for Everton like he was for us at the World Cup," said the Manchester City front-man.
"In the qualification he wasn't there at all, then at the World Cup he played three games and in those games he showed his quality.
"I think the Premier League will see another great player from Bosnia."
Besic, who became his country's youngest senior international with a debut aged 18, played in each of his nation’s World Cup group fixtures, but could do little to prevent a premature group stage exit.
Nonetheless, this glowing reference from his international teammate will be a fantastic confidence boost as he prepares to make his debut for the Toffees.
Besic will wear the number 17 shirt for the club, as he takes over a shirt previously worn by Anders Limpar, Andrei Kanchelskis and Tim Cahill.
He could make his first appearance for Roberto Martinez's men in Leon Osman's testimonial match against FC Porto on Sunday.