The first minister says he wants to "open a dialogue" with the new UK government, soon after the general election, about how justice is delivered in Wales in future.
Mark Drakeford will chair a committee to oversee the Welsh Government's own response to an independent commission.
People in Wales were being "let down by the current system", it concluded.
The commission was chaired by Lord Thomas of Cwmgiedd, once the most senior judge in England and Wales.
Mr Drakeford, making the first official response to last month's conclusions, described it as a "watershed moment, with a "treasure trove" of evidence and advice.
But he said the report also revealed "real and insurmountable practical challenges" which came from the division of responsibilities between Westminster and Wales - while it outlined challenges to the Welsh Government and said that it could do more.
Examples included funding legal apprenticeships, creating a Welsh law council to promote legal education and investment in technology.
£1.165bnspending on justice system a year in Wales
£370spending per person
£723mUK govt spending, a real terms drop by a third since 2009-10
£14mestimated cost of 200 staff if justice policy devolved
Mr Drakeford said the new cabinet justice committee would be tasked with taking forward the recommendations which fall to the Welsh Government and overseeing discussions with the new UK government.
He also promised an assembly debate on the commission's findings in the new year.
The Ministry of Justice's initial response last month was that devolution of justice would be too costly and lead to significant duplication.
Lord Thomas said: "I'm sure when they look at the logic of it, think of the position of justice in society, the savings, what can be done in a small nation and how much better it could be, I very much hope the argument will win them round."