It replaces the disability living allowance for children and recipients of this benefit will be transferred over in phases before spring 2023.
For the first time, people will be able to apply for the new benefit online, as well as by phone, post or in person.
The child disability payment helps families with extra costs a disabled child may have.
Its consists of two components, one for care and one for mobility, each of which have three different levels of payment.
Social Security Scotland is rolling out the devolved benefit after trials in Dundee City, Perth & Kinross and the Western Isles.
Ben Macpherson, Minister for Social Security, said: "I'm really pleased that child disability payment is now available to eligible families all over Scotland.
"This is a significant milestone in the introduction of our new social security system.
"Families who find themselves in need of support will be able to access this help in a way that suits them best."
He continued: "In the months and years ahead thousands of families will benefit from our simplified and much less stressful system, which will treat everyone with dignity, fairness and respect.
"Social security is a shared investment in building a fairer Scotland and we encourage those who are eligible for support to apply."
Scottish Green MSP Maggie Chapman also encouraged families to apply, saying: "This new Scottish benefit can provide much needed support to families with children with a disability, and was successfully trialled in Dundee and other areas."