Although exact terms of the settlement are confidential it is thought to be the largest sum of compensation ever publicly disclosed for failure on a government IT contract.
However, EDS has made no formal acceptance of blame for the overpayment, saying only that it settled over "issues with the IT system".
A statement from HMRC claimed that all parties had successfully concluded " an aggregate settlement of £71.25m including an upfront payment and payments of additional amounts over time".
An investigation by the National Audit Office (NAO) found that the tax credit system had led to overpayments estimated at £2.2bn, of which almost £1bn is likely to be written off by HMRC. A total of £460m was also overpaid because of fraud and error.
The NAO report stated: "During the introduction of tax credits there were very serious problems which affected the stability, speed and availability of the computer system which delayed the processing of claims and led to incorrect payments being made."
The system failures led to thousands of low-income families having to pay back sums of money to the taxman, in some cases up to £1,500.
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NatWest outage comes a day after Barclays' IT systems shut out customers and staff