The former chancellor said his plan, if he becomes prime minister, involves keeping any one-off borrowing to an “absolute minimum” by seeking “efficiency savings” across Government departments.
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.
Mr 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 his leadership rival, Liz Truss, has been forced on the defensive on the issue.
Foreign Secretary Ms Truss suggested at the weekend that there would be no “handouts” 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.
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.
Mr Sunak has labelled Ms 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, Mr 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.”
With the Bank of England forecasting inflation is set to hit 13% – with average household energy bills predicted to reached almost £4,000 – Mr Sunak, writing in The Sun, said his rival’s measures “won’t touch the sides”.
Tory former Northern Ireland secretary Brandon Lewis, who is backing Ms Truss, earlier said they would look to do “whatever we can” to help people under pressure from rising inflation.
He told BBC Radio 4’s Today programme: “We will look to do whatever we can to help people – that’s what an emergency budget is about.
“She’s willing to do more to help people but her focus is around doing it in a way that puts more money in people’s pockets, creating a high-growth economy with higher wages, more people in work.
“So rather than having handouts, what we do is have a low-tax economy that’s driving growth and therefore with people having more money in their pockets, they’re better placed to deal with some of the challenges that we see.”
Mr Lewis said it was still possible deliver tax cuts while dealing with inflation.
“We want to do both, want to make sure we’re getting on top of inflation and you can – to get on top of inflation whilst still putting more money in people’s pockets,” he told Times Radio.
“I think it’s a false premise to argue that you can’t deal with inflation whilst making sure that people are better off at the same time.”