The First Minister hailed the move as the "boldest and most ambitious anti-poverty measure anywhere in the UK".
She said more than 400,000 children and their families would be eligible by the end of 2022.
Charities welcomed the commitment, with John Dickie, director of the Child Poverty Action Group, insisting it would be “a real lifeline for the families across Scotland”.
Ms Sturgeon made the announcement in a speech on the final day of the SNP's online conference.
She said it would involve "hard choices" elsewhere in the Scottish Budget.
But she said this was a choice the SNP, together with their junior partners in government in the Scottish Greens, had decided to make.
The Scottish Child Payment goes towards the costs of supporting a family and was set up following the devolution of new welfare powers to Scotland.
The benefit, unique in the UK to Scotland, currently pays out to the families of more than 105,000 children under the age of six, but is being expanded to all under-16s by the end of next year.
Ms Sturgeon said: "I am very pleased – indeed proud – to announce today that our budget on December 9 will fund the doubling of the Scottish Child Payment immediately from the start of the new financial year.
"The Scottish Child Payment will increase to £20 per child per week – four times the amount originally demanded by campaigners – from April.
"That means the doubled payments will reach over 100,000 children under age six in just four months’ time.
"And when we extend the Scottish Child Payment to all under-16s at the end of next year, over 400,000 children and their families will be eligible.
"This is, without doubt, the boldest and most ambitious anti-poverty measure anywhere in the UK."
Peter Kelly, director of the Poverty Alliance, said doubling the payment would help “loosen the grip of poverty on the lives of thousands of children in Scotland”.
Scottish Labour deputy leader Jackie Baillie said the increase to the payment needed to go further, calling for it to be upped to £40 a week.