The 17-year-old broke into the first team at the Championship club during the second half of the last campaign and is rated highly at Home Park.
Gosling can play in both defence and midfield and has been involved with the England Under-17 and Under-18 sides. He has scored twice in 23 appearances for the south coast club.
"He is one for the future," said Toffees boss David Moyes after sealing a deal which is believed to be worth as much as £1.5million.
Meanwhile, the Toffees have also allowed striker Lukas Jutkiewicz to join Plymouth on loan.
The teenage striker joined the Goodison Park club from Swindon Town in the summer.