Deputy First Minister John Swinney told MSPs at Holyrood that applications are now being accepted from those who have a terminal illness or are aged 70 and over.
The scheme is open to those who were abused while in care, including those who attended private boarding schools.
However, Mr Swinney said payments would not be made to those who were sent to boarding schools by their parents but only those placed there by the state and where the institution was acting in loco parentis.
Applicants will not be required to submit evidence of having been abused, but will require documentary evidence which shows they were in care.
The initiative is only the precursor to a larger statutory scheme which will offer redress to all those who suffered abuse in care, and which could cost ten of millions of pounds.
Mr Swinney said: “We wholeheartedly accept the need to provide acknowledgement and tangible recognition of the harm done to children who were abused in care in Scotland, while acknowledging that such recognition cannot in any way take away the pain that individuals have suffered.
“We are all too aware that, because of their age or health, some survivors may not live long enough to apply to the statutory scheme.
“Survivors asked us to develop an application process which is as straightforward as possible, whilst making the scheme robust and credible. This is what we have designed.”
Mr Swinney said the £10,000 payment was broadly in line with interim awards made by redress schemes looked at in Australia, Canada and Ireland.
He said the Scottish Government would bear the entire cost, with an initial £10m set aside in the current financial year.
The government is currently looking at asking organisations responsible for the abuse of children in care to contribute to the larger scheme, he said.
Survivors of child abuse, many now in old age, have been calling on the Scottish Government to consider the issue of financial redress for many years, but stepped up pressure following the establishment of the Scottish Child Abuse Inquiry in 2015.
Mr Swinney said he hoped survivors would receive their payment within a month of applying.
However, he said it was “impossible to predict” how many people would come forward.