More than 500 people are to be offered the payments beginning on 1 July, which the Welsh government said it hopes will help them "on a path to live, healthy, happy and fulfilling lives."
A care leaver is someone who has spent time in foster or residental care, away from their family, but is ready to leave because they are turning 18.
The trial, named the Basic Income pilot scheme, will cost £20 million, last for three years and be evaluated carefully to examine its effect on those who are participating in it, Welsh ministers said.
The Labour-run government said it would provide independence and security to those who have faced immense challenges during their childhood, giving them greater control over their lives and empower them to make decisions about their future.
Tiff Evans of Voices from Care Cymru, speaking on behalf of young people who have experienced care, welcome the scheme. She said: “This is a brilliant opportunity for care leavers in Wales.
"It is good to see that care leavers in Wales are being thought of and Welsh government are providing this opportunity for them as young people to become responsible, control some parts of their lives and have a chance to thrive and be financially independent."
She added: “We thank Welsh government for investing in them and their future and we look forward to other changes and developments for care experienced young people in Wales in order for them to reach life aspirations.”
Care leavers taking part will also receive individual advice and support to help them manage their finances and develop their financial and budgeting skills.
Local authorities will play a key role in supporting them throughout the pilot.
Voices from Care Cymru will also work with the young people to give them advice on wellbeing, education, employment and help them plan their future after the pilot.
Jane Hutt, minister for social justice, said: “We will carefully evaluate the lessons learnt from the pilot. Listening to everyone who takes part will be crucial in determining the success of this globally ambitious project.
"We will examine whether Basic Income is an efficient way to support society’s most vulnerable and not only benefit the individual, but wider society too.”
Joel James, the Conservative Party shadow social justice minister, said the scheme could well “create more problems than it solves”.
"It's been proven time and again that so-called Universal Basic Income doesn't work,” he said “Look at Finland, who ditched their scheme after two years in favour of a new scheme that encouraged people to actually take up employment or training."
"We recognise that this is a vulnerable group and they need extra support, but this is completely the wrong way to go about it. It's typical Labour, but it's obvious that giving out free money won't be a quick fix."
Mark Drakeford, Wales first minister, said: “Many of those involved in this pilot don’t have the support lots of people – myself included - have been lucky enough to enjoy as we started out on our path to adulthood.
“Our radical initiative will not only improve the lives of those taking part in the pilot, but will reap rewards for the rest of Welsh society.
"If we succeed in what we are attempting today this will be just the first step in what could be a journey that benefits generations to come.”
The Basic Income pilot scheme will be limited to care leavers who reach their 18th birthday between 1 July 2022 and 30 June 2023.
Participants in the pilot can choose whether to receive their payment either monthly or two times a month, paid to them by an external provider and taxed at the basic rate of tax at source.