The museum, at Shambellie House near New Abbey, Dumfries, is the smallest of Scotland's five national museums.
It costs £220,000 a year to run the attraction, which receives just 10,000 visitors a year, with 5,000 people taking in the shop, cafe and grounds.
The museum, which is currently shut for winter, will not now reopen this spring.
National Museums Scotland said it, like many other bodies, had had to work with reduced amounts of public cash.
The organisation has been making efficiency savings, has reduced staffing numbers and is working to attract donations and sponsorship, but the financial climate is still said to be difficult.
Bruce Minto, chairman of the Trustees of National Museums Scotland, said closing the National Museum of Costume was a "necessary part of a range of cost savings".
He stated: "It is with great regret that we have taken the decision to close the National Museum of Costume.
"While this has been an extremely difficult decision for the Board of Trustees, we are clear that in the current challenging financial climate it is a necessary part of a range of cost savings which ensures the long-term future of our national collections."
Dr Gordon Rintoul, director of National Museums Scotland, said the "difficult decision" to shut the museum had been made "after extensive consideration of all other options across National Museums".
He added: "The low number of visitors to the site along with the high operational costs is simply not sustainable.
"In addition, the domestic layout of Shambellie House places limitations on it being used effectively as a national museum."
National Museums Scotland also includes the National Museum of Scotland, the National War Museum, the National Museum of Rural Life and the National Museum of Flight.
The National Museum of Costume attracted just 0.6% of all total visits to the national museums.
Shambellie House was built in 1856 by the Stewart family, who gifted a costume collection to National Museums Scotland, which has maintained the building since 1978 and operated the museum there since 1982.
It is now planned to display a selection of items from the costume collection in the new art and design galleries, which are scheduled to open in 2016 at the National Museum of Scotland in Edinburgh.
Dr Rintoul also said National Museums Scotland "remain committed to delivering services to Dumfries and Galloway" by loaning items, staging exhibitions and outreach programmes.
"We believe this could provide greater access to our national collections across the whole region and reach a higher audience of both locals and tourists alike," he said.
South of Scotland MSP Jim Hume has campaigned against the closure plans since they were first consulted on last year.
The Liberal Democrat said he was "disappointed" but not surprised by the decision.
"It was clear from my meetings with officials last year that this was always the preferred option and no amount of campaigning and lobbying from those of us keen to keep Shambellie House open was going to sway decision makers in Edinburgh," he said.
"There is obviously no good time to close Shambellie House, but to close it with immediate effect during the winter shutdown displays particularly poor timing. The local community, and tourists, should have one last opportunity to visit the exhibits.
"At the very least, National Museums Scotland must allow for a temporary reprieve to ensure that people have a final opportunity to visit this culturally important museum before it is lost to people forever."
Mr Hume said after the National Museum of Scotland in Edinburgh underwent a substantial programme of investment, people in Dumfries and Galloway would "understandably question why Shambellie House has had to be axed".
He urged National Museums Scotland to keep its pledge to hold more events in Dumfries and Galloway, saying the organisation should "publish full details of their roving exhibitions and outreach programmes as soon as possible".
He also called for Culture Secretary Fiona Hyslop to set out her plans for the site.
Mr Hume said: "Now that Shambellie House will revert to Scottish Government ownership, the Culture Secretary must come forward with concrete plans on her intentions for the site to ensure it is protected for future generations and not sold off to the highest bidder."