The clubs agreed a deal for the Bury academy graduate in November and he became a Toffees player when the transfer window reopened on Saturday.
Foulds signed his first professional contract with League One Bury in February and made two first-team appearances, both in cup competitions.
He has also played on a trial basis for Everton's Under-18 and Under-21 sides.
Everton have completed the signing of 17-year-old Bury defender Matthew Foulds for an undisclosed fee.