He called on Prime Minister Theresa May to accept all refugee children with a legal right to be in the UK, ahead of French plans to shut the "Jungle" camp.
Mr Jones said Wales was ready to support the UK government in welcoming greater numbers of vulnerable children.
The Home Office said its priority was to ensure the safety of children in the Calais camp.
Mr Jones made the case in a joint statement with Kirsty Davies-Warner, head of Oxfam Cymru; Mary Powell-Chandler, head of Save the Children in Wales; Huw Thomas, head of Christian Aid Wales, and Jim Barnaville, co-chair of Citizens Cymru Wales.
The statement said Mr Jones had written to the prime minister expressing "our shared view that the UK government should take urgent action to resettle more unaccompanied refugee children in the UK and offering the support of Wales in welcoming them".
With the French government set to demolish the Calais "Jungle" camp, the statement asked Mrs May to "allow all those refugee children with a legal right to be in the UK into the country with the utmost urgency".
"The children in the camp, some of whom have been as young as eight years old, have fled conflict and persecution and are now stuck in northern France, deeply traumatised and at considerable risk," the statement read.
"Every extra day we take places these children at greater risk."
The Immigration Act included an amendment, originally put forward by Lord Dubs, that places a legal requirement on government to arrange for the transfer to the UK of unaccompanied refugee children from Europe.
The statement welcomed a commitment from the UK government to resettle children who have a legal right to be in the UK, but called on Ms May to do more, adding: "Wales is ready to support the UK government in welcoming more vulnerable children as refugees via the Dubs amendment".
The children's charity Unicef has appealed to the UK government to speed up the transfer of unaccompanied child refugees in Calais.
It has said it is concerned the planned closure of the camp will lead to children there disappearing before they are processed.
A Home Office spokesman said: "Our priority must be to ensure the safety and security of the children in the Calais camp."
He said that Home Secretary Amber Rudd had told the French Interior Minister Bernard Cazeneuve that the UK government intended to transfer as many minors as possible eligible under the Dublin Regulation before the start of the clearance.
Under the Dublin Regulation, asylum seekers must make an initial asylum claim in the first EU state they reach, but can apply for asylum in other EU states if they have family there.
"In addition, children who are eligible to come to the UK under the Dubs Amendment to the Immigration Act 2016 must be looked after in safe facilities where their best interests are properly considered," the spokesman added.
"Work is continuing on both sides of the Channel to ensure this happens as a matter of urgency."
First Minister Carwyn Jones has joined the charity sector in calling for urgent action to help child refugees.