New rules brought in last year require councils to support children who stay on with their foster parents.
Children and Families Minister Edward Timpson said the scheme had already been a "spectacular success".
Council leaders said they were committed to the changes but warned of a funding shortfall.
The government says 2,300 young people between the ages of 18 and 21 now remain with their foster families.
It says this is a big improvement because young people leaving foster homes often miss out on the sort of help parents provide to children aged over 18.
Cllr David Simmonds, deputy chairman of the Local Government Association, said: "We are absolutely committed to ensuring as many of our most vulnerable young people as possible can stay with their foster families if they choose to, but these changes have to be sufficiently resourced."
He added: "Councils remain concerned that the cost to councils of keeping children with foster parents until they are 21 has been significantly underestimated by government."
The rule change, which came into force in March 2014, requires local authorities to provide financial support for every young person wanting to stay with their foster families until their 21st birthday.
The government is providing funding towards the cost of the Staying Put scheme.
Mr Timpson added: "We're already seeing spectacular success and we've made available new money, £44m, to councils over the next three years to try and meet that demand.
"And of course we'll need to keep that under review to make sure that every young person who chooses, because it's their choice, with the support of their foster carer, to remain living with them, gets that opportunity to do so.
"Because we know what a huge difference it can make to their prospects and future life."
Government figures for 2014 showed that 41% of 19-year-olds who left care were not in education, employment or training, compared with 13% of all 19-year-olds.
How do I work the oven?
Heidi, 18, from Stoke, said she would not have known where to start if she had had to set up home on her own.
"I'd be like how do I work the oven, how do I work the cooker?" she said.
"I haven't had to worry that when I turned 18 I would have to find myself a place of my own, or leaving home.
"I've been able to concentrate on college work and not worry about moving out," she added.
Heather Clegg, who has fostered over 50 children, said: "It's very difficult for young people at 18 to move out into the world with everything you have to do to get a flat and find work... the prospect of that succeeding is very low."
According to The Fostering Network charity some 78% of children in care are placed with foster families.
The Fostering Network's chief executive Kevin Williams said the recent legislation was proving extremely effective.
But he called on the government to provide more support, guidance and finances to ensure "even more young people can benefit from this remarkable change in the care system".
He told BBC Radio 5 live: "Remaining in a loving, caring foster home enables them to move into adulthood in a stable way and we believe it will then reduce some of the poorer outcomes for looked-after children - young people who get involved in the criminal justice system, young people who may end up homeless, suffering with mental health issues.
"So, a real improvement in staying with their foster carers helps those young people move into independence at a more natural level, at a more appropriate time in the way that their peers do."