The charity said research it commissioned found that 40 per cent of teachers were "not confident" they would be able to identify a young carer in their class.
Meanwhile, more than a third (34 per cent) of teachers surveyed thought there were young carers at their school who were not sufficiently supported and 29 per cent said they didn't think their school had any particular ways of supporting young carers.
Under the Care Act 2014 and the Children and Families Act 2014, teachers and other professionals working with children have a statutory duty to identify young carers and refer them to the local authority to be assessed for support.
The government had been due to publish a cross-government strategy to support carers over the summer but it is yet to appear. Speaking in March, former children's minister Edward Timpson said the Department for Education was "exploring various policy avenues" to help local authorities, schools and professionals improve the identification and support of young carers and their families.
Barnardo's chief executive Javed Khan described the current situation as "simply not acceptable".
"Austerity has meant local authorities have had to cut back on adult social care and the result is children are picking up the pieces," he said.
"A quarter of the children supported by Barnardo's young carers' services are carrying out more than 30 hours a week of caring - that's the equivalent of a full-time job.
"It's clear from our research that there is a lack of awareness among teachers that needs urgently addressing. Schools need to take more responsibility to make sure young carers are properly supported.
"Looking after their family members is something that our young carers are incredibly proud of but it shouldn't be at the expense of their childhoods or their futures."
The National Association of Head Teachers (NAHT) said many families where a child has caring responsibilities may also be on low incomes, and children from these families should be automatically registered for the Pupil Premium.
"[Pupil Premium] funding is specifically targeted at students who face challenges not unlike those faced by young carers," Paul Whiteman, NAHT general secretary said.
"They should get this support automatically, particularly as the person they are caring for may not be able to apply for the money themselves. The additional funding would help schools to properly resource the ways in which they can identify and help young carers.
"Supporting young carers' mental health is another concern, but funding has not kept pace with this demand. The government's green paper on mental health this week includes some welcome proposals but NAHT has been pressing the government for some time to take a more rounded approach to mental health provision, particularly to take some of the emphasis away from schools and re-assert the importance of well-resourced and accessible local support services."