He confirmed an additional £12 million has been found to take the fund up to the £50 million needed to offset the UK Government welfare reform policy.
Mr Swinney announced his decision as MSPs prepared to vote on the SNP's spending plans for the year ahead.
"I want to be clear to the chamber that none of the funding I have announced here is a solution to the bedroom tax," he said.
"This is about mitigation - about picking up the pieces from Westminster's iniquitous policy.
"The only real solution is to scrap the bedroom tax altogether, and this Government believes the only way to do that is for this Parliament to have the full powers over welfare in Scotland."
The SNP and Labour set aside wider political differences to find a solution to the problem during negotiations on the Budget Bill.
Mr Swinney said UK Department for Work and Pensions (DWP) rules mean the Scottish Government is constrained in what it can legally spend.
The DWP set a cap at £22.85 million for discretionary housing payments, which can be used to support tenants at risk of falling behind in rent and getting into debt.
The SNP administration said the total support that can be provided directly by councils to individuals is £38 million, still short of the £50 million required.
Deputy First Minister Nicola Sturgeon has written to the department asking for the cap to be removed, he said.
Mr Swinney said: "I give Parliament the assurance today that if the DWP says no, the Scottish Government will put in place a scheme to make this additional £12 million available to social landlords so that we need not see any evictions in Scotland this year as a result solely of the bedroom tax."
Mr Swinney said there is an "enormous" challenge to mitigate all UK welfare reforms.
The Scottish Government will spend more than £244 million alleviating their impact between 2013-14 and 2015-16, at the expense of spending in other areas.
But he singled out the controversial Tory-Lib Dem policy, called a tax by opponents, which cuts the amount of benefit people can get if they are deemed to have a spare room.
"The bedroom tax is an iniquitous and damaging policy," he claimed.
His Budget Bill also confirms another year of the council tax freeze, support for free university tuition, free prescriptions and free personal care.
From next January, the SNP administration will fund free school meals for all schoolchildren in the first three years of primary school.
Labour finance spokesman Iain Gray welcomed the announcement on the "bedroom tax" and said there are already ways round the rules.
Money could be given to councils and housing associations to cover rent arrears, he said.
But he accepted it is better to help people avoid getting into debt in the first place.
His party put forward an amendment to the Budget Bill, calling for MSPs to ensure no tenant will face eviction because of the benefit cut.
Mr Gray said: "We are almost there. The full £50 million funding is in place and if the DWP restriction remains then a workable alternative exists and our amendment today endorses that position."
He added: "When this budget was introduced I called it a 'don't rock the boat, do nothing, get through the referendum' budget.
"If we pass my amendment we will have turned it into a 'sink the bedroom tax boat to the bottom of the sea' budget."
Conservative finance spokesman Gavin Brown said the extra money would be better spent boosting economic growth and employment.
"Our party's position has been absolutely consistent all the way down the line: we believe that where funds are available it should be invested into the economy," he said.
Labour's Jackie Baillie said the SNP had performed a welcome, "spectacular U-turn" over the so-called bedroom tax.
"Let's be clear that by voting for this budget today, we will effectively bring an end to the bedroom tax in Scotland," she said.
Ms Baillie brought forward a Bill earlier this year with measures to help protect tenants affected by the housing benefit reform.
"For more than a year Labour has been calling for the Scottish Government to fully mitigate the bedroom tax," she said.
"We stood shoulder to shoulder with campaigners across Scotland."
She added: "The money pledged by the Finance Secretary today to end the bedroom tax in Scotland is absolutely and unreservedly welcome. This removes the need for any social landlord to take eviction proceedings due to arrears of bedroom tax.
"It is a strength of government, not a weakness, to listen, then to act and to admit when you are wrong."
Liberal Democrat leader Willie Rennie welcomed additional free childcare, the extension of other free school meals and extra funding for colleges.
He said: "This is not a budget that would be delivered if (the Liberal Democrats) were a majority group in this Parliament, but it is good enough for us to support.
"We recognise the position we are in, and we recognise the engagement by John Swinney and the improvements we have made.
"This is a budget that I think is good enough, and it is good enough because John Swinney has engaged in the process."