Video report by ITV News Economics Editor Joel Hills
Mr Sunak, in an update to MPs, confirmed that employees can continue claiming up to 80% of their salaries if they are unable to work after England's lockdown ends on December 2.
And the next self-employed income support grant - which covers November to January - will increase from 55% to 80% of average profits, capped at £7,500.
ITV News Political Editor Robert Peston has more analysis on the UK's Covid crisis:
He said the government's highest priority remains "to protect jobs and livelihoods".
Under the furlough scheme, which pays up to £2,500 of wages, employers will only have to cover National Insurance and pension contributions.
The scheme will be reviewed in January to "decide whether economic circumstances are improving enough to ask employers to contribute more".
The scheme, which was set up in March when the UK was put under its first coronavirus lockdown, was first extended throughout November owing to the second national lockdown in England, before being extended again for another five months on Thursday.
Mr Sunak guaranteed furlough funding for Scotland, Wales and Northern Ireland.
The "upfront guaranteed funding" for devolved administrations has been increased from £14 billion to £16 billion, the chancellor announced.
"This Treasury is, has been and will always be, the Treasury for the whole of the United Kingdom," he added.
Wales First Minister Mark Drakeford welcomed the cash but suggested it should have been announced sooner.
"At long last, businesses and workers have some certainty about this vital support scheme to help them get through the pandemic," he said.
Mr Sunak said he was extending the scheme past the end of England's lockdown because "it’s clear the economic effects are much longer lasting for businesses than the duration of any restrictions, which is why we have decided to go further with our support".
He added: "Extending furlough and increasing our support for the self-employed will protect millions of jobs and give people and businesses the certainty they need over what will be a difficult winter."
The Job Retention Bonus, which paid firms £1,000 for bringing staff back after furlough, will end in January and a new retention incentive will be announced "at the appropriate time", Mr Sunak said.
Watch Rishi Sunak's furlough extension announcement in full:
Boris Johnson's plan is to revert back to the tiered system of local lockdowns, which vary across England, and businesses impacted by the continued restrictions will be able to furlough their staff and have the government pay 80% of their unworked wages.
Businesses will have flexibility to use the scheme for employees for any amount of time and shift pattern, including furloughing staff full-time.
The update comes after an announcement that the Bank of England will pump another £150 billion into the UK's battered economy as England's second lockdown threatens to damage it even further.
Speaking at a press conference after announcing the increase in quantitative easing, Bank of England Governor Andrew Bailey said: “We believe there is value in acting quickly and strongly to support the economy and avoid the risks of any short-term disruption.”
Mr Bailey said the Bank’s work into negative interest rates is continuing, but stressed it stands ready to take more action when needed.
Mr Bailey added: “We are here to do everything we can to support the people of this country – and we’ll do it and will do it quickly.”
Shadow chancellor Anneliese Dodds said the chancellor is "always a step behind".
She attacked Mr Sunak for waiting "until the last possible moment after jobs have been lost and businesses have gone bust" before announcing support.
She added: "Now when the lockdown was announced, the prime minister said furlough would be extended for a month - five hours before that scheme was due to end.
"Two days later, realising the self-employed had been forgotten, there was a last-minute change to the self-employed scheme. And now, further changes.
"The chancellor's fourth version of his winter economy plan in just six weeks. The Chancellor can change his mind at the last minute, but businesses can't. "
She welcomed the new funding but said "questions remain" over the support package.
"How many jobs could have been saved if this Government had recognised reality and let businesses plan for the future?" she asked.
England's second lockdown, as with first, has forced the closure of all non-essential shops, hospitality, leisure and entertainment venues.
People are advised to stay at home and should only go out for essential reasons.
All indoor cross-household socialising is banned and people can only meet outdoors with a maximum of one person from another household.
Cash grants of up to £3,000 per month for businesses forced to close under Tier 3 restrictions are still available on top of the support announced on Thursday in the Commons.
The chancellor said he would "leave it to the people" to decide whether the Government's actions were right.