A report published by the Scottish Children's Reporter Administration (SCRA) revealed that 3,275 children were referred on the grounds that they have, or are likely to have, a close connection to someone who has carried out domestic abuse.
More than one-third of those children were aged under four at the time.
However, the number of annual referrals is likely to rise as the new provision only came into force in June 2013. Prior to this, children at risk from domestic abuse were referred on more general care and protection grounds.
If concerns are serious enough, children affected by domestic abuse could be referred to a children's hearing. This could result in a demand that parents get help, or even leave a violent partner, with possible options including removing the child to stay with another family member or a foster carer if they are still at risk.
Tam Baillie, Scotland's Commissioner for Children and Young People, said: "Over time, this provision will help make children who live with domestic abuse more visible and give us a better picture as to how the Children's Hearings System supports them and their parent or carer.
"We still have some way to go to ensure that all those working to support children and young people affected by domestic abuse have the right knowledge, skills and experience to effectively support them, keep them safe, ensure their well-being and protect their rights."
To meet the "close connection" criteria, the child must either live in the same household as someone who has carried out domestic abuse or be in "significant contact'"with such an offender. These provisions are aimed at covering not just immediate family but also the partner of a child's parent who does not live in the same home.
In response to the figures, welfare groups have called for better measures to protect Scotland's youngsters before they suffer from violence or distress. Matt Forde, head of NSPCC services in Scotland, argues that domestic abuse is a major underpinning factor in most cases of disruptive behaviour in children.
He said: "Research shows a child's development can be harmed by witnessing physical or emotional abuse at a young age.
"Children as young as five from violent or abusive homes are two to four times more likely to hit, slap or push other children; pick on others or; break, damage or destroy someone else's belongings."
The children affected need prompt support, Mr Forde added: "Reaching children before they begin to act out their trauma is the only way to break the downward spiral of violence and give Scotland's next generation a safer and happier future."
A spokesman for the SCRA said: "Exposure to domestic abuse can have a profound impact on children and young people. We will continue to work with our partners in the Children's Hearings System to ensure the children and young people whose lives are blighted by this issue receive appropriate support and intervention."
A Scottish Government spokeswoman added: "Domestic abuse can have a devastating effect on children and is rightly condemned by all. We support a range of work to help families affected.."
Children or young people affected by domestic abuse can contact ChildLine on 0800 1111 or visit www.childline.org.uk. Anyone who has concerns about a child's welfare can call ParentLine Scotland on 08000 28 22 33.