Allegations against the Rt Rev George Bell were first made by the victim in 1995 but were not investigated or referred to the police.
Bell was Bishop of Chichester from 1929 until his death in October 1958.
Current Bishop Dr Martin Warner issued a formal apology after the Diocese of Chichester settled the civil claim.
He paid tribute to the victim's courage in coming forward to report the abuse.
"I am committed to ensuring that the past is handled with honesty and transparency," he said.
The allegations of sexual offences against Bell date from the late 1940s and early 1950s and concern an individual who was at the time a young child.
The victim first reported the abuse to the then Bishop of Chichester, Eric Kemp, in August 1995 but no action was taken until contact was made with Archbishop of Canterbury Justin Welby's office in 2013.
Following settlement of the claim, Bishop Warner wrote to the survivor formally apologising and expressing his deep sorrow.
The victim's solicitor said they remained bitter their 1995 complaint was not properly listened to or dealt with.
"The failure to respond properly was very damaging, and combined with the abuse that was suffered has had a profound effect on my client's life," said Tracey Emmott.
"For my client, the compensation finally received does not change anything."
Another retired Church of England bishop was jailed last week for a string of offences against teenagers and young men.
Peter Ball, 83, the former Bishop of Lewes and Bishop of Gloucester was sentenced to 32 months for misconduct in public office and 15 months for indecent assaults, to run concurrently.

The communities and local government committee said the eight biggest firms built more than half of all new homes.
MPs said the government should do more for smaller builders who do not have the scale to bid for big projects.
But the Home Builders Federation, which represents large and small businesses, said only big firms could spread the risks large-scale projects pose.
The committee also said local authorities should prepare land for home building.
That would include providing the infrastructure needed, such as roads and public transport.
"The housing market is broken, we are simply not building enough homes," said Clive Betts MP, chair of the committee.
"Smaller builders are in decline and the sector is over-reliant on an alarmingly small number of high-volume developers, driven by commercial self-interest and with little incentive to build any quicker.
"If we are to build the homes that the country so desperately needs, for sale and for rent, then this dominance must end."
Government promises more affordable homes
Labour promises to build one million new homes
The committee found that smaller builders struggled to obtained land for development, as local authorities focused on large sites which only big companies could afford to take on.
The Homebuilders Federation said: "We fully support the committee's call for measures to assist smaller builders, encourage new entrants and scale up specialist housing sectors, such as the retirement housing market.
"The vast majority of the big increases in housing supply in recent years have come from the larger, mainstream builders - but we need more builders of all sizes and specialisms if we are to tackle our acute housing shortage."
In February the government promised to build more affordable houses and help people buy and rent, after admitting the current market was broken.
The housing strategy for England included giving councils powers to pressure developers into starting building on land they own.
Ministers also pledged to make renting more "family-friendly" with longer tenancies offered.
However, Labour called the measures announced "feeble beyond belief".
A victim who was sexually abused as a young child by a former Bishop of Chichester who died in the 1950s has received compensation from the Church.
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MPs have called for an end to the dominance of big home-building firms to fix the "broken" housing market.