In his draft budget the Finance Secretary said he was finding the money from other departments to give cash to councils for social tenants in difficulty.
Mr Swinney rejected a call from Labour to provide £50m next year for a no evictions policy helping protect tenants and social landlords.
His budget also included extra cash for house building, money for additional childcare places and for further education colleges.
Mr Swinney said the Scottish Government had to take cash from devolved services to mitigate UK welfare reforms. The Scottish Welfare Fund will be maintained for another two years costing £33m a year.
He said: "This government will use the resources we have, to invest £68m in each of the next two years to limit the damage of Westminster's welfare cuts."
He said he agreed with the United Nations the 'bedroom tax' should be abolished and would press UK ministers to scrap it.
He said: "We do not have legal or financial powers over welfare benefits, but neither are we a government that will walk by on the other side."
He said the £20m in this current financial year would be for councils to increase the Discretionary Housing Payments for "those struggling the most with the costs of the bedroom tax".
Labour said it offered nothing for tenants next year. Iain Gray, Labour's Finance Spokesman said it would only see councils having to prioritise between the deserving and undeserving victims of the 'bedroom tax'.
Mr Swinney, however, said the blame for any hardship lies with the Conservatives at Westminster.
He said: "We have taken action to deal with this within our powers. There's no provision in 2014/15 budget as I have no intention of letting the Westminster Government off the hook."
Mr Gray said: "In parliament John Swinney refused to commit to helping struggling Scots beyond this year because, he said, he doesn't want to let the Westminster government 'off the hook'.
"So now we know the poorest, most vulnerable, disadvantaged Scots will be left on John Swinney's hook just so he can make a political point. That's not good enough."
The UK Government gave cash to councils to help tenants in transition, but it was derided as insufficient.
Glasgow received more than £3m from the fund, but officials said the level of need was closer to £10m and the cash would either run out in a few months or people would be turned away.
The Scottish Government cash will allow councils to top up the fund which they are allowed to do by up to an additional 150%, in Glasgow's case an extra £4.5m
Shelter Scotland had proposed the measure in its Banish the Bedroom Tax campaign.
Graeme Brown, chief executive, said: "This is a victory not only for supporters of Shelter Scotland's campaign, but for the people suffering hardship who will benefit from this move.
"We hope that local authorities across Scotland will act quickly to top up their discretionary housing payments budget so that the maximum number of people this year can be helped."
The £3.5bn budget is split between 10 departments with the biggest share going to health which accounts for almost £3.2bn.
Mr Swinney also announced an increase of £111m for affordable housing over three years and another £390m the following year.
He said £190m over two years would help fund an additional 125 hours of childcare for three and four-year-olds and looked after two year olds, which he said was worth £7000 to a family.