Home Secretary Amber Rudd has announced the child protection expert will become the fourth person to lead the probe after the resignation of Dame Lowell Goddard last week.
Professor Jay previously led an inquiry that revealed at least 1,400 children were subjected to sexual exploitation in Rotherham.
The Church of England has already sent more than 7,000 documents to the inquiry, all of which were said to be relevant.
The investigation is looking into whether the Church and public bodies have taken seriously their duty of care to protect children from sexual abuse in England and Wales.
A separate inquiry is being held in Scotland.
Amber Rudd said: "The Independent Inquiry has a vital role to play in exposing the failure of public bodies and other major organisations to prevent systematic child sexual abuse.
"I'm delighted Professor Alexis Jay has agreed to chair the Inquiry. She has a strong track record in uncovering the truth and I have no doubt she will run this independent Inquiry with vigour, compassion and courage.
"Let there be no doubt; our commitment to this Inquiry is undiminished.
"We owe it to victims and survivors to confront the appalling reality of how children were let down by the very people who were charged to protect them and to learn from the mistakes of the past."
Prof Jay was already heavily involved in the inquiry as a member of the four person panel that assisted the chair.
She said: "I am committed to ensuring this Inquiry does everything it has set out to do and does so with pace, with confidence and with clarity.
"Be in no doubt - the Inquiry is open for business and people are busier than ever working hard to increase momentum."