The clubs had already agreed an undisclosed fee for the 20-year-old.
Derry said on their website: "Michael will be joining the development squad at Celtic Park under manager Stevie Frail with a hope of making the step to the senior team as soon as possible."
Duffy had impressed during two trial periods at Celtic Park.
The Northern Ireland Under-21 international recently signed a two-year contract extension with Derry, who finished eighth in the 2014 League of Ireland Premier Division.
"His departure will be a blow to the Candystripes, although the fee received could help manager Peter Hutton bring in some new faces ahead of the 2015 campaign," added Derry.
The league season in the Republic of Ireland runs from March to October.
Derry City winger Michael Duffy has completed his move to Celtic after passing a medical and agreeing personal terms with the Scottish champions.