Westminster's Scottish Affairs Committee has urged both administrations to expedite the necessary procedures which will enable the Scottish Government to set the cap on Discretionary Housing Payments (DHPs).
The payments have been used to help those facing financial difficulty as a result of controversial changes to housing benefit, where people lose cash if they are deemed to have more rooms than they need.
About £50 million is needed to fully mitigate the loss in housing benefit payments for those council and housing association tenants who have been affected.
The UK Government offered to transfer the power to set the cap on DHPs north of the border to the Scottish Parliament earlier this month, with Scotland Office Minister David Mundell contacting Deputy First Minister Nicola Sturgeon on the issue.
MPs on the Scottish Affairs Committee said they would now "urge both governments to expedite the necessary procedures which will enable the Scottish Government to lift the cap on DHPs in Scotland as quickly as possible".
Committee chairman Ian Davidson said the decision to transfer the power to Holyrood showed that "devolution works".
The Labour MP added: "We now urge the UK and the Scottish governments to move quickly to enable the Scottish Government to lift the cap on DHPs in Scotland as soon as possible."
Ms Sturgeon has already said that local councils should "plan on the basis that all losses of housing benefit incurred by social tenants due to the 'bedroom tax' can be fully mitigated".
The Department for Work and Pensions will give Scottish councils £15 million this year for DHPs.