The UK Government finally agreed to a request by the Scottish Government that would allow it to give enough cash to ensure no-one loses out as a result of the benefit cut if social housing tenants have a spare room.
It means Discretionary Housing Payments can now be paid to all tenants to allow them to pay their rent.
The Scottish Government will now be able to allocate the £50m it pledged in the next year to nullify the impact of the bedroom tax in Scotland.
Scotland Office Minister David Mundell announced the deal, which has been welcomed by the Scottish Government and housing campaigners.
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."
Nicola Sturgeon, deputy First Minister, said it had taken too long but now urged all affected tenants to apply to their council for help.
She said: "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.
"We will never turn our back on people in need, and I am pleased to finally be able to get on and help people.
"We have been pressing since January for Iain Duncan Smith to remove this cap and at last Westminster has seen sense."
Housing campaigners said the move was a welcome step until the policy was scrapped, but feared considerable damage had already been done.
Graeme Brown, director of Shelter, 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.
"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 as a result of the bedroom tax."