The scheme was due to expire at the end of March but the extension suggests the Government believes the economy will still be struggling to cope with the impact of the coronavirus pandemic.
Mr Sunak also extended Government loan schemes until the end of March to help support businesses.
The Chancellor said: "Our package of support for businesses and workers continues to be one of the most generous and effective in the world - helping our economy to recover and protecting livelihoods across the country.
"We know the premium businesses place on certainty, so it is right that we enable them to plan ahead regardless of the path the virus takes, which is why we're providing certainty and clarity by extending this support, as well as implementing our plan for jobs."
Mr Sunak also announced that the next Budget will be on March 3, 2021.
Under the furlough scheme the Government will continue to pay 80% of the salary of employees for hours not worked until the end of April.
Employers will only be required to pay wages, National Insurance Contributions (NICS) and pensions for hours worked; and NICS and pensions for hours not worked.
The business loans schemes had been due to close at the end of January.
The schemes, which will now stay open until the end of March are: the Bounce Back Loan Scheme, Coronavirus Business Interruption Loan Scheme, and the Coronavirus Large Business Interruption Loan Scheme.