The Labour-led Welsh Government insist that lives will be protected by the policy, meaning the NHS in Wales could save an extra £92 million a year as a result.
It is predicting the change would save up to 100 lives and 20,000 casualties in the first decade.
Not all 30mph roads will see their speed limits reduced as councils have the power to exempt certain roads from the scheme.
The project is costing around £33 million to implement and has proven controversial, with reports of the new 20mph signs being defaced in areas including Conwy, Gwynedd, Newport, Torfaen, Wrexham and Flintshire.
Drakeford told Senedd members the cut in residential speed limits would save lives.
“That is why this Government will stick fast to the decision that we have made, endorsed twice on the floor of the Senedd, not by a simple majority, but by a super majority, because of the human stories that lie behind the decision we have made and will go on implementing here in Wales,” he said.