"I hope this decision makes clear how seriously the Church of England understands the need for change and our profound commitment to creating a safer church," Mr Welby said.
"As I step down I do so in sorrow with all victims and survivors of abuse," he added.
A spokesperson for Prime Minister Keir Starmer said he "respects the decision that has been taken and his thoughts remain first and foremost with all the victims”.
Last week, an independent report found inaction from the Church represented a "missed opportunity" to bring Smyth to justice before his 2018 death.
In his resignation statement, Mr Welby said he was "told that police had been notified" at the time and that he "believed wrongly that an appropriate resolution would follow".
He also spoke of his "profound sense of shame at the historic safeguarding failures" of the Church over the days since the report was published.
"For nearly twelve years I have struggled to introduce improvements. It is for others to judge what has been done," he said.
The Archbishop of Canterbury is the head of the Church of England and leads 85 million Anglicans in 165 countries around the world.
Archbishop of York Stephen Cottrell said the Church had made "real progress" in safeguarding under Mr Welby's leadership but added: "There is much further to go."
The Church's lead safeguarding bishop, Joanne Grenfell, said the archbishop's resignation "does not absolve any of us from bringing about the wholesale changes in culture and leadership that are essential".
Former vicar Mark Stibbe, a survivor of Smyth's abuse, said Mr Welby had "done the right thing" in resigning.
"What I think the survivor group would like is more resignations because that means more accountability," he told Channel 4 News.
