Eligible farmers who have lost more than 25% of their April and May income will be entitled to up to £10,000, to cover 70% of their lost income.
A similar scheme has already been announced in England.
Farmers’ leaders have welcomed the funding but there has been criticism from the Conservatives and Plaid Cymru.
Ms Griffiths said the “closure of the food service sector has had an immediate and significant impact” on the dairy sector with a need to “support those farms hardest hit by a situation which is primarily outside of their control”.
She added: “I am, therefore, pleased to confirm dairy farmers in Wales will be eligible for support helping them adapt to the exceptional market conditions and ensuring they can continue to operate without impacting animal welfare and the environment.”
Further details of the scheme have yet to be announced.
NFU Cymru deputy president Aled Jones said the loss of the food service market was causing “severe financial hardship”.
“We are pleased that Welsh Government has recognised the impact this is having on many dairy farming businesses and has now come forward with a hardship package for those impacted most by the current market conditions,” he said.
But Plaid Cymru’s rural affairs spokesman Llyr Gruffydd said the scheme could have been announced weeks ago and added: “When I pressed Lesley Griffiths on the need for support in a Senedd committee on Thursday she claimed her department had been working on proposals for weeks.
“It nevertheless took an announcement for England for her to finally act. The scheme she has adopted also seems to be a carbon copy of the one announced by Defra.”
The Welsh Conservatives’ rural affairs spokesman Andrew RT Davies said the announcement was “better late than never”, adding that Ms Griffiths “needs to ensure the money gets out quickly as possible with many farmers badly hit”.
“The absence of key detail in this announcement is a concern and clarity is required ASAP,” he added.