The former chancellor said his plan, if he becomes prime minister, involves keeping any one-off borrowing to an “absolute minimum” by pressing Whitehall departments to make savings.
His team said the approach would aim to replicate previous measures used to fund support for Ukraine.
This resulted in departments and devolved administrations being asked to find underspends from their capital budgets, which involves money spent on investment and things used to create future growth.
Sunak sought to keep the focus on cost of living amid renewed concerns over energy bills increasing further in the coming months and at a time when Truss has been forced on the defensive on the issue.
Last week, the Bank of England announced projections that could see inflation rise above 13% in October, while another projection found that the energy price cap could rise to £3,358 annually from October, and could hit £3,615 from January.
Foreign secretary Truss suggested at the weekend that there would be no support for hard-pressed families if she won the race for No 10, and that her priority was reducing the tax burden.
But her allies have insisted she is committed to helping families struggling with soaring bills.
Party leader Ed Davey said the government should cover the shortfall to energy suppliers to ensure they can supply customers at the current rates.
The party estimated the cost of the policy would be £36 billion and suggested the windfall tax on oil and gas company profits should be expanded to help cover it.
The two Tory leadership hopefuls will continue campaigning on Tuesday before a hustings session with party members in Darlington, a so-called “red wall” area turned blue under Boris Johnson’s leadership in 2019.
Sunak has labelled Truss’s plan for tax cuts in an emergency budget as a “big bung” for large businesses and the better off, adding it would do little to help those most in need over the coming winter.
On his own plans, Sunak said: “This winter is going to be extremely tough for families up and down the country, and there is no doubt in my mind that more support will be needed.
“In spring I set out a series of measures to provide help, with the most support targeted at the most vulnerable. But bills are going up by more than anyone expected and the next government will need to act.
“As chancellor I put in place a framework to support hard-working families and pensioners to bring bills down. People need proven methods that will deliver for them quickly. So I will use the framework I created to provide further support and give millions of people the peace of mind they desperately need ahead of the winter.
“I’m very clear about what is required to help people, and as soon as we know how much bills will go up by, I will act.
“And it’s important for people to know how this extra support will be paid for. In order to keep any one-off borrowing to an absolute minimum I will first seek efficiency savings across Whitehall to provide direct support for families to help with the unprecedented situation we face.”
A Truss campaign source said: “The question for Rishi on the economics of his new handout pledges is: how is he going to fund these new promises?
“Three weeks ago he was saying more borrowing was irresponsible and inflationary. Has he changed his mind? It’s feels like another big U-turn.
“How can Rishi’s borrowing not be inflationary, but Liz’s tax cuts are? Intellectually it’s as watertight as a sieve.”
It comes as Scotland’s first minister Nicola Sturgeon called for an urgent meeting between the heads of the devolved administrations to address the cost-of-living crisis.
Meanwhile, Tony Danker, director general of the Confederation of British Industry, warned the government that it must have “all hands to the pump” as former prime minister Gordon Brown also called for swift intervention to address rising energy costs.
But Downing Street said Johnson had no plans to introduce major new fiscal measures before the end of his premiership.