Tosun scored 41 goals in 96 appearances for Besiktas since the start of 2014-15 season.
The 26-year old, who had been in London awaiting permission to complete the move, said he was eager to be involved in Everton's next Premier League game, against Tottenham on Saturday 13 January.
"I want to play as soon as possible", he said.
"It feels great to be an Everton player", Tosun told Everton's website.
The fact Tosun has been content to turn his back on playing in the knockout phase of Europe's premier club competition underlines the financial and sporting lure of the Premier League.
"I bring my skills here and everybody is going to see what kind of player I am", he said.
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Tosun signed a four-and-a-half-year deal, with the move coming just hours after Everton confirmed the transfer of Ross Barkley to Chelsea.
Tosun has helped the Turkish side reach the last 16 of the Champions League, where they will meet Bayern Munich. I hope they can make a lot of assists for me!
Everton's ultra-defensive approach to the recent Premier League draw at Anfield was roundly criticised, not least by his Liverpool counterpart, Jürgen Klopp, and Allardyce admits improvement will be necessary to achieve a first win across Stanley Park since 1999. He has got some good movement, he's two-footed, he's not the biggest but he's efficient in the air in the box.
He has given a number of younger players a chance ahead of the Netherlands global too, so if a bid comes in for the 24-year-old, the Toffees may look to recoup as much of the £23.6 million as they can and admit he was a signing that failed.
Tosun was born and raised in Germany and played for their youth worldwide teams but chose to switch to Turkey and has 25 appearances for their senior side, including at Euro 2016.