The former Prime Minister made an intervention after commissioning a report which criticised the current levels of support amid rising costs.
It found the flat-rate payments offered by the government will fall at least £1,600 short of making up for recent changes to living costs and benefits faced by a couple with two children.
Now the report is calling for new measures to bridge the shortfall in family finances, for benefits to be upgraded in line with inflation, and for debt deductions from Universal Credit to be paused.
Others endorsing it include the Trussell Trust, the Joseph Rowntree Foundation, the Child Poverty Action Group as well as the Methodist Church, the Bishop of Durham, the Hindu Council UK and the Muslim Council of Britain.
It has also been endorsed by Welsh First Minister Mark Drakeford, Metro Mayors, Tracy Brabin of West Yorkshire, Andy Burnham of Greater Manchester, and Sadiq Khan of Greater London.
Mr Brown said: “This is a crisis that Britain hasn’t seen in decades.
"We need targeted support for families on the lowest-incomes, not just cuts in taxes or flat rate payments which don’t account for the specific needs of people on the brink of poverty.
“There should be no argument that a permanent increase in Universal Credit is the only way to take a sure step towards a solution.
"This crisis goes far beyond politics; this is a moral issue - our responsibilities to our neighbours and in particular to those who have the least and whose needs are the greatest.
“The incoming Prime Minister has a moral responsibility to ensure that everyone has enough to live on, through this crisis and beyond.
"We cannot be at ease when millions are ill at ease and cannot rest content as long as there is so much discontent. Our society will be stronger when we help the weak and will be richer when we help the poor."
Revd Graham Thompson, President of the Methodist Church in Britain, who have endorsed the report, said: “Churches and other faith groups are on the front line of offering support to families who are already being swept under by rising costs.
“We know that millions of families aren’t simply making hard choices between heating and eating, but are having to go without both completely.
“If people aren’t give enough support to live, we don’t dare to imagine what will happen this winter. The government now have a duty to step up and take firm and long-lasting action to ensure not only that this crisis doesn’t deepen, but that it doesn’t happen again.”
A UK government spokesperson said: “We understand that people are struggling with rising prices which is why we have acted to protect the 8 million most vulnerable British families through at least £1,200 of direct payments this year with additional support for pensioners and those claiming disability benefits.
“Through our £37bn support package we are also saving the typical employee over £330 a year through a tax cut in July, allowing people on Universal Credit to keep £1,000 more of what they earn and cutting fuel duty by 5p saving a typical family £100.”