The striker is married to Ece Tosun, who he tied the knot with in 2016.
This deal makes Tosun the most expensive football player in Turkish Super League history.
The 26-year-old, who scored 41 goals in 96 games for Besiktas since 2014, has penned a four-and-a-half year contract at Goodison Park.
"This lad looks like he is strong mentally and has a good pedigree", Allardyce told Everton's official website of his new recruit.
Tosun also underlined that he is looking forward to proving himself to Everton fans and showing them just what he is made of.
"I bring my skills here and everybody is going to see what kind of player I am". "He is Turkey's No.1 striker and has scored goals in the Champions League and the Turkish League but he is desperate to play here in the Premier League".
'He said he really wanted me here and to feel that, of course, is another big reason to come to Everton. Also I can use both feet the same and I believe I am a good striker in the box.
Everton seals £27m Cenk Tosun deal
The Tottenham game will be the first game I am available for and I'm very excited about it already'.
"There would have to be some exits, I don't think there is any doubt about that because we've got about 30 players", the manager said in the Liverpool Echo.
All being well, he should make his Premier League bow next weekend as Allardyce's side visit Wembley Stadium for a hard meeting with Spurs.
With Lukaku gone, Everton have failed to score in eight Premier League games this season - and have not had a shot on target in their past two home games.
Striker Dominic Calvert-Lewin has taken more shots (37) than any other Everton player so far this season, with a 50% success rate.