Francis' move followed mounting pressure from Australian priests and even the prime minister, Malcolm Turnbull.
Adelaide Archbishop Philip Wilson was convicted in May of failing to report to police the repeated abuse of two altar boys by a paedophile priest in the Hunter Valley region north of Sydney during the 1970s.
He became the highest-ranking Catholic cleric ever convicted in a criminal court of covering up abuse.
Wilson had declined to resign pending an appeal of his case.
Mr Turnbull added his voice to those calling for his sacking in an appeal to Francis on July 19.
The Vatican said that Francis has accepted Wilson's resignation.
It is the second major announcement of a sex abuse-related resignation in as many days, after Francis' dramatic sanctioning this weekend of a US cardinal, suggesting he is keen to clean up before he heads to Dublin next month for a Catholic family rally.
Wilson, who denied the accusations, had immediately stepped aside after he was convicted but refused to resign pending an appeal. Francis has appointed a temporary administrator to run the diocese in the meantime.
In a statement issued by the archdiocese, Wilson said he had submitted his resignation to Francis of his own will on July 20 - a day after Mr Turnbull's call - and said he hoped his decision would help abuse victims and the rest of the Catholic community heal.
He said: "I had hoped to defer this decision until after the appeal process had been completed. However, there is just too much pain and distress being caused by my maintaining the office of archbishop."
Wilson was sentenced by the Newcastle court to 12 months in detention.
Francis' decision to accept the resignation is significant given he has previously refrained from taking action against accused bishops that might be perceived as prejudicing outcomes in civil or criminal cases.
Another Australian prelate, Cardinal George Pell, for example, has been on leave as the Vatican's finance czar while he faces criminal trial on accusations of sexual abuse.
But Pell, who denies the charges, remains a cardinal, head of the Vatican's economy secretariat and a member of Francis' core group of nine cardinal advisers.
Francis, though, is under increasing pressure to sanction bishops who have abused, botched handling abuse cases or otherwise covered them up. There are calls for a full-fledged church investigation in the United States, and criminal probes are under way in Chile as the next phase of the abuse scandal - accountability for bishops who failed to protect their flocks from abusive priests - is gaining momentum.
In the United States, bishops and cardinals are coming under fire for failing to reveal what they knew and when about the abuse of adult seminarians and minors allegedly committed by Theodore McCarrick, the former archbishop of Washington.
Francis on Saturday accepted McCarrick's resignation as cardinal, and imposed on him unprecedented penalties for a cardinal even before his canonical trial is completed, including living a lifetime of penance and prayer and living isolated from others.
Stay up to date with the latest news stories from a Christian perspective. Sign up to our daily newsletter and receive more stories like this straight to your inbox every morning.
