The Croatia international impressed Everton boss Ronald Koeman when his side travelled to Hajduk Split in the Europa League qualifier earlier this month and Vlasic left Croatia's U21 squad on Thursday to finalise the move to Goodison Park.
The 19-year-old described his move to the Premier League as "the opportunity of a lifetime".
He said: "I am very excited about the opportunity to play for this big club.
"The Premier League is the best league in the world and this is one of the top clubs in England. It is such a big honour for me to be here."
Vlasic aims to help Ronald Koeman and his new team-mates into the Champions League and believes the Merseyside club can match his ambitions.
He added: "When there is a manager like Mr Koeman wants you, there is no need to talk about it much.
"I wanted to come here because of the manager, because of the big players, because of the reputation of the club. It is the opportunity of a lifetime.
"The ambition of the club is to ultimately play in the Champions League and I want to help achieve that goal. I believe in myself and in this team - I think we can do great things together."
Koeman called his new signing "one of the big talents in Europe", and added: "Nikola is one of those players who can play a big part in Everton's future and in what we want to achieve as a club."