More than 70,000 Scottish households are due to benefit to the tune of a total of about £50 million a year.
The move has been welcomed by charities and campaigners who warn the charge is forcing vulnerable families into poverty.
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Last night, Deputy First Minister Nicola Sturgeon said she was "delighted that in future anyone who has been affected by this unfair policy will receive the help they need".
But she hit out at Conservative Work and Pensions Secretary Iain Duncan Smith, accusing him of dragging his feet on the issue since January.
Under the policy, council or housing association tenants deemed to have a spare bedroom have seen their housing benefit cut by 14%.
Both the SNP and Labour have called on the Coalition Government to scrap the bedroom tax entirely.
The issue has become central to the attempts of both sides to win over less-affluent voters in the independence campaign.
The SNP say only independence can ensure the charge is scrapped forever.,
However, Labour say they can abolish the tax sooner, if the party is returned to power at Westminster in 2015.
The levy was designed to bring the rules for council houses into line with those for private accommodation and free up larger homes for those on waiting lists.
Critics say the policy was always fatally flawed because there are not enough one or two-bedroom properties for tenants to move to.
Scottish ministers have been pressing the Coalition for months to relax rules limiting the amount of compensation tenants can receive directly.
UK ministers have now agreed that a cap on discretionary housing payments can be set by Scottish ministers - effectively allowing them to spend as much as they want.
The change will have to be approved by both Westminster and Holyrood in the coming months.
However, Coalition sources said it would cover all of the current financial year, which began last month.
Last night, 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."
She added: "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."
Labour MSP Jackie Baillie said: "This iniquitous tax should never have been introduced. The bedroom tax needs axed."
That call was echoed by Graeme Brown, director of Shelter Scotland, who described yesterday's announcement as great news for tens of thousands of households.
He added: "This is not only a victory for common sense, it is a victory for social justice."
Mary Taylor, chief executive of the Scottish Federation of Housing Associations, also criticised the Coalition, saying the decision had taken a long time.
She said her organisation was pleased Holyrood would now effectively have the power to ensure tenants were no longer forced to pay the tax and added: "We remain concerned about the arrears backlog that has been caused in the previous financial year (2013/14) and the cost impacts upon landlords."
Keith Dryburgh, policy manager at Citizens Advice Scotland (CAS), said: "Our evidence has shown that the bedroom tax has had a devastating impact on people in Scotland, including some of the most vulnerable groups in our society."
CAS says that in the first nine months of the policy, local authority rent arrears issues across Scotland increased by 34%.
Announcing the new power for Holyrood, Conservative Scotland Office Minister David Mundell said: "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."