Holyrood offered powers to offset housing benefit changes in Scotland
The UK government is to devolve the power to set the cap on Discretionary Housing Payments (DHP) in Scotland.
The payments have been used by the Scottish government to mitigate the impact of housing benefit changes.
Scottish ministers had called for a cap to be lifted on the amount of DHP payments it was able to make.
Deputy First Minister Nicola Sturgeon welcomed the offer. She said it would help people affected by what she called the "bedroom tax" in Scotland.
The move would enable the Scottish government to "provide vital assistance to thousands of hard-pressed Scots", she added.
DHPs can be used by local authorities across the UK to provide additional funding for people in receipt of housing benefit who need extra support.
At present each local authority must operate within a formula-based spending cap set by the Department for Work and Pensions.
As part of his budget in February, Scottish Finance Secretary John Swinney announced extra funding for discretionary housing payments in Scotland to help social housing tenants whose benefits are cut because they have a spare room.
UK ministers have argued they have removed a "spare room subsidy" from tenants, but the changes have been dubbed the "bedroom tax" by critics.
Mr Swinney said data from the UK government, which controls housing benefits, showed Scottish ministers were permitted to allocate about £38m.
But this fell short of the £50m total he said was needed to fully mitigate the effects of the removal of the spare room subsidy, which affects an estimated 76,000 Scottish households.
Mr Swinney had asked the UK government to lift the cap.
The UK government has now proposed giving the Scottish government the flexibility to pass on more funding from its existing block grant to local authorities.
It would be up to the Scottish government and local authorities how they chose to allocate the money. This would mean that the Scottish government would have the power to set the DHP cap for Scottish local authorities in future.
Scotland Office Minister David Mundell said the power would be transferred to the Scottish government through a Section 63 Order, which would require the agreement of the UK and Scottish governments before being approved by both the UK and Scottish Parliaments.
Mr Mundell said: "I have completed a programme of visits to all Scottish local authorities and believe that transferring this power to the Scottish government is the correct thing to do.
"The UK government believes in taking a pragmatic approach to devolution and we believe in a United Kingdom that gives Scotland the best of both worlds. I hope that officials from both governments will now be able to take this forward."
He has written to Scottish Deputy First Minister Nicola Sturgeon outlining the proposal.
An adviser to Liberal Democrat ministers in the UK coalition government said the plan was "another example of Liberal Democrats delivering for devolution in Scotland".
Ms Sturgeon welcomed the move and said Scottish ministers would "work to ensure the law is changed as quickly as possible".
She said: "We had already set aside the money to be able to help every household in Scotland affected by the 'bedroom tax' - once we have the powers, we will be able to use it and provide vital assistance to thousands of hard-pressed Scots.
"I am delighted that in future anyone who has been affected by this unfair policy will receive the help they need and would encourage them to contact their local authority to apply for assistance through the DHP scheme.
She added: "The DHP scheme is the only legal way - under the powers that Scotland currently has - to provide regular financial payments to people on housing benefit. But the only way to get rid of the 'bedroom tax' for good is through the powers of an independent Scottish Parliament.
"We know that Scots want welfare decisions to be made and taken by the Scottish Parliament. The 'bedroom tax' has been rejected by people right across Scotland, yet is still being imposed on us by the UK government."
Scottish Labour's welfare spokeswoman, Jackie Baillie, said: "The Scottish and UK governments have so far been more interested in taking each other on, rather than come to a solution which works.
"Now that we finally have an agreed means of effectively ending the impact of the 'bedroom tax' in Scotland, the Scottish government must now act and give councils and housing associations clear and unequivocal assurances over DHP payments.
"Labour is committed to scrap the tax if we win the general election next year. The best way of ending the 'bedroom tax' forever is to elect Ed Miliband in 2015."
Mary Taylor, chief executive of the Scottish Federation of Housing Associations, said: "We are pleased to see that Holyrood will now effectively have power to ease the worry of the 'bedroom tax' for social housing tenants in Scotland.
"However, we remain concerned about the arrears backlog that has been caused in the previous financial year and the cost impacts upon landlords."
Graeme Brown, director of the housing charity Shelter Scotland, said: "This is not only a victory for common sense it is a victory for social justice."