The Scotland international had looked set to join Coventry, only for Everton to pip the Sky Blues to the signature of the talented 21-year-old.
As revealed on Thursday by planetfootball.com, the Merseysiders made their late move for the player in order to soften the blow of losing Alessandro Pistone through injury.
"We`re pleased to get him," Smith told the official Everton website. "I think he is a player with great potential.
"He has worked very hard to get from the youth team all the way through to senior level.
"So we are looking forward to getting a player who is 21 years old and who looks to have a very bright future."
Naysmith added: "I`m delighted to have made the move to such a big club as Everton."
The Toffees also announced another new arrival - former striker Graeme Sharp, who scored in the 1984 FA Cup final, has returned to Goodison as Fans` Liason Officer.
Click here to listen to Smith and Naysmith`s interviews with the Everton website
Click here for our story breaking news of Everton`s late bid for Naysmith
