The Scottish Government has repeatedly called for the rules to be changed to allow more money to be spent on Discretionary Housing Payments (DHPs) - which have been used to help those facing financial difficulty as a result of the so-called "bedroom tax".
Deputy First Minister Nicola Sturgeon said "at last Westminster has seen sense" on the issue as she pledged to work to ensure the change is made as quickly as possible.
At the moment, the amount that can be spent on DHPs in Scotland this year is limited to about £38 million while the Scottish Government has said about £50 million is needed to help those affected by the controversial housing benefit change.
Scotland Office Minister David Mundell has written to Ms Sturgeon, offering to transfer the power to set the cap on DHPs north of the border.
The UK Government believes the move shows its commitment to devolution, with Mr Mundell describing it as the "correct thing to do".
Transferring the power will mean the Scottish Government will be able to pass on more funding from its block grant to local councils, which distribute payments under the scheme.
Ms Sturgeon 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: "We will never turn our back on people in need, and I am pleased to finally be able to get on and help people.
"But the fact is that this decision has taken far too long. We have been pressing since January for Iain Duncan Smith to remove this cap - and at last Westminster has seen sense and have given us what we requested.
"We will now work to ensure the law is changed as quickly as possible.
"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."
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."
For the power to set the DHP cap for Scottish local authorities to be devolved, a Section 63 order needs to be agreed by both the UK and Scottish governments, before being approved by the UK Parliament and Holyrood.
An adviser to Liberal Democrat ministers in the coalition government said the move was "another example of Liberal Democrats delivering for devolution in Scotland".
The adviser added: "I hope we can work together to see this power devolved because it would be another power the Scottish Parliament would get."
Labour's Jackie Baillie said the move provided an "agreed means of effectively ending the impact of the bedroom tax in Scotland".
She demanded: "The Scottish Government must now act and give councils and housing associations clear and unequivocal assurances over DHP payments.
"We've had enough dithering. For more than a month, Scottish councils have had no idea about what was happening to support our poorest and most vulnerable. Now clarity has been provided, let's get on and put the money in place."
Mary Taylor, chief executive of the Scottish Federation of Housing Associations (SFHA), said: "We are pleased the Scotland Office has taken a pragmatic approach, even though this decision has taken a long time.
"The additional monies are required to increase the Discretionary Housing Payment (DHP) fund to £50 million for this financial year and fully mitigate the impact of the 'bedroom tax' in Scotland.
"We hope that today's announcement from the Scotland Office will bring an end to many weeks of uncertainty for Scottish housing associations and their tenants."
She added: "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 (2013-14) and the cost impacts upon landlords.
"We still wish to see the 'bedroom tax' repealed by the UK Government, so that tenants in other parts of the UK can be relieved of this iniquitous burden.
"Our hope is that these new powers will at least allow the Scottish Government to alleviate the impact of the 'bedroom tax' in Scotland."
Graeme Brown, director of the housing charity Shelter Scotland, said: "This is great news for the more than 70,000 households affected by the iniquitous bedroom tax in Scotland and we congratulate the Scottish Government for making it happen.
"Ultimately the only way to banish the bedroom tax is to scrap the policy for good.
"Until that time, the Scottish Government can now fulfil its commitment to help all those struggling to pay their rent as a result of the bedroom tax. This is not only a victory for common sense it is a victory for social justice."