In a wide-ranging interview for BBC Radio 4, Rt Rev Dame Sarah Mullally also spoke of the support she has received from women after her appointment was announced, and her opposition to assisted dying legislation.
Bishop Sarah said: “The Church of England has had significant safeguarding failures, for which we are truly sorry.
“The Church of England has made progress over the last ten years, certainly in the time that I've been a bishop.
“The voice of survivors is much more prevalent and involved. There are much more rigorous processes. The professionalism and the number of safeguarding professionals are much higher.
“We can never be complacent, and it certainly will be something that I will pay primary attention to as I become Archbishop of Canterbury.”
The Archbishop-elect spoke of the role that independent scrutiny can play in the church’s safeguarding procedures, with audits of dioceses underway.
As reported by Premier Christian News, the church is reviewing a complaint against Bishop Sarah over her handling of an abuse case during her time in the Diocese of London.
Survivors of abuse have told Premier that they have been re-traumatised by the way their cases were handled after coming forward, despite the church’s repeated claims to have learned from past failures.
Bishop Sarah was speaking to former Prime Minister Theresa May, who was guest editing the Today programme on Wednesday.
She will be the first woman to hold the position, which has led to some opposition in the Anglican communion.
“One of the wonderful things over the last couple of months is I've had hundreds and hundreds of letters of support,” Bishop Sarah said, "I had a wonderful letter from an ordained woman who said the announcement of my appointment was the first time that she felt affirmed as a priest.”
Responding to the opposition, Bishop Sarah said she always encourages clergy, even if they are against the ordination of women.
The Archbishop-elect also reaffirmed her opposition to assisted dying to “protect the most vulnerable in society”. Legislation is currently being debated in the House of Lords, where the bishops sit.
“We need to have a view where everybody is made in the image of God.”
“Those who support the bill talk about choice. I'm not sure we have choice - we don't properly fund palliative care, we don't properly fund palliative-care research,” she added, referencing her previous role as chief nursing officer.
In a divided society, she said the church can be “providing spaces for listening and for increased community cohesion,” adding that she has witnessed a “hunger for relationships” among young people coming to church for the first time.