Deputy First Minister Nicola Sturgeon has written to the UK Government asking for rules to be changed so the money can be spent on discretionary housing payments (DHPs).
She said she is "more than willing" to spend the cash, which would take the amount of help available in Scotland to a total of £50 million in 2014/15.
Scottish Labour's Iain Gray welcomed the move, which he described as a U-turn by ministers.
In a letter to UK Welfare Minister Lord Freud, Ms Sturgeon said increasing the current DHP funds from £35 million to £50 million in 2014/15 would help the 76,000 "hard-pressed" people in Scotland hit by the so-called bedroom tax.
She said the Scottish Government is currently spending "up to the legal limit" allowed under Department of Work and Pensions (DWP) rules and called for the coalition at Westminster to lift a cap on the amount of money Holyrood can provide.
Speaking today, Ms Sturgeon said: "The bedroom tax penalises some of the most vulnerable people in our society. We know that over 12,000 children are affected by the bedroom tax and 80% of households hit contain an adult with a recognised disability.
"We have already provided as much help as legally possible to those suffering from this unjust policy but we are unfairly restricted in what we can do. For example, despite Scotland having 20,000 more households affected by the bedroom tax than London, the DHP allocation for Scotland in 2014/15 is £35 million less than London.
"The Scottish Government is currently spending up to the legal limit in order to mitigate the effects of the bedroom tax on thousands of people across Scotland. We are more than willing to put in the extra £15 million, which would increase the amount of help available to a total of £50 million.
"If Westminster lifts the legal cap - which they can easily do - we will be able to help the 76,000 people in Scotland who are suffering from this cut.
"In order to make this legally possible Westminster needs to lift the cap for Scotland and UK ministers should act now."
Ms Sturgeon also said that in order to abolish the bedroom tax, ministers need the powers that would come from independence.
She told BBC Radio Scotland's Good Morning Scotland programme: "I think it's absurd that we're having to try to jump through hoops, trying to find money within a fixed budget to mitigate the impact of a policy that's opposed by a majority of the Scottish Parliament, opposed by a majority of the Scottish people, being imposed on us by a Government we didn't vote for.
"It would make much more sense to have the powers over welfare in Scotland so that we didn't have a bedroom tax in the first place, but we want to do the right thing."
The DWP rejects the "bedroom tax" tag and says the reality is that "a spare room subsidy" has been removed from social sector tenants.
Scottish ministers have already agreed to contribute £20 million each year to the DHP budget, which is also getting £15 million from the UK Government, making a £35 million pot.
An extra £15 million in Wednesday's budget would take the total to £50 million, thought to represent the total cost of the "bedroom tax" in Scotland.
Responding to the Scottish Government's announcement, Labour's finance spokesman Mr Gray told Good Morning Scotland: "We've been asking the Scottish Government for many months to find this money, to make it available, so that no-one in Scotland need pay the bedroom tax, so of course we welcome the money.
"We do look for the commitment that it will be available come what may. Last week we presented (Finance Secretary) John Swinney with a way in which this can be done if DWP won't agree to lift the cap.
"It's been a long journey. Back in September John Swinney said he wouldn't do this because it was Westminster's fault, not his. He then came up with about half of what was needed, £20 million, and said he couldn't possibly go any further.
"He now has gone further, so it is a U-turn, but it is a very welcome one indeed."
He went on: "I suppose if there was a difference between Nicola (Sturgeon) and myself it is that I want to end the iniquity of the bedroom tax here in Scotland, as she does too, but I want to see it ended in London and Liverpool and Birmingham and Manchester as well, which is exactly what a Labour government will do next year."
Scottish Trades Union Congress (STUC) general secretary Grahame Smith said: "The bedroom tax has been a campaigning priority for STUC since the announcement of the measure in 2012. It is testament to the campaigning activities of a whole range of community organisations that we have achieved some respite for those affected by the tax.
"STUC also recognises that both the Scottish Government and the Scottish Labour Party have consistently opposed the bedroom tax, even if they have hitherto not seen eye to eye on the tactics required to eliminate it. We are delighted that agreement now appears to have been reached.
"Today's success cannot hide the fact that a tranche of welfare reform measures and public service cuts continue to bite deeply into the living standards of our people and our communities. STUC will continue to campaign alongside Scottish communities to address the serious challenges we face."
Citizens Advice Scotland (CAB) policy manager Keith Dryburgh said: "Across Scotland, CAB advisers are seeing lots of people every week who are being hit by the bedroom tax. In the evidence we have published so far we have highlighted the impact on specific vulnerable groups like disabled people, families with separated parents and people in rural areas who find it hard to move to alternative accommodation.
"We believe that these groups should be exempted from the policy altogether. But in the meantime anything that can be done to mitigate the impact of the policy on these groups is to be welcomed. It's important that governments work together - and with local councils, landlords and charities as well - to help people avoid falling into crisis."
Gordon MacRae, head of communications and policy at Shelter Scotland, said: "The bedroom tax not only makes it harder for sick and low income households to pay their rent - forcing thousands into debt or out of their home - but it also poisons the very fabric of Scotland's social housing system."
He added: "We were delighted that the Scottish Government listened to our call to top up the discretionary housing payments and we welcome today's announcement of a further £15 million if agreement can be reached with the Department for Work and Pensions to protect tenants' current levels of benefit."
A DWP spokeswoman said: "The Scottish Government's call for the current 150% cap on discretionary housing payment top-ups to be increased doesn't fit with our experience of how DHPs are currently working in Scotland.
"The UK Government set aside £20 million of additional DHP funding support which local authorities could apply for through a bidding scheme. After operating for four months the scheme closes today and so far only 11 [around a third] Scottish authorities have made a bid for additional funding."