The Scottish Child Payment was expanded to include all eligible families with children under the age of 16, as well as increasing the payment from £20 per week to £25.
Poverty campaigners described the shift as a “watershed moment”, with so many people trying to apply the website reportedly crashed on Monday morning.
But Nicola Sturgeon told the PA news agency as she visited a school in Glasgow: “There’s massive demand, and I actually welcome the fact there has been such a strong demand for applications for this extended benefit today.
“Yes, that’s created some difficulties for the website today, which will be sorted over the course of the day.”
By early afternoon on Monday, the website had been restored.
A spokeswoman for Social Security Scotland said: “The introduction or expansion of services is never without risks. We had staff on hand to monitor issues and to fix these quickly. This will continue throughout this week as we expect to see continued levels of high demand.”
The First Minister, describing the payment as a product of the Scottish Government’s priority to lift “children and young people out of poverty”, stressed the importance of the policy.
“In my view, this is one of, if not the most, important policy decisions the Scottish Government has taken in the last decade,” she said.
“Poverty robs young people of their life chances and their potential and any government that wants the best for its young people but also the best for the country overall, will invest in order to tackle that,” she added.
Some 400,000 children in Scotland are now thought to be eligible for the Scottish Child Payment, and the Joseph Rowntree Foundation said if the scheme was replicated across the rest of the UK, 5.3 million youngsters in England, Wales and Northern Ireland could benefit.
Chris Birt, associate director for the social change organisation in Scotland, said the benefit should show the UK Government that “prioritising those on low incomes is possible”.
It also shows other devolved Governments that the “constraints” on their powers are “no barrier to compassionate and significant action to support families”, he added.
It means eligible families will receive £1,300 a year per child – which Mr Birt described as a “welcome boost” at a time when many household budgets are “stretched to breaking point”.
He said: “The full rollout of the Scottish Child Payment is a watershed moment for tackling poverty in Scotland, and the rest of the UK should take notice.
“At £1,300 per child, per year it will be a welcome boost to family budgets that are stretched to breaking point already.
“No child should live in poverty so there is clearly more to do, but the Scottish Government should be commended for prioritising spend on this vital measure at this time.
“But this is not just a cost-of-living crisis measure, it is an enduring investment in our children.”
The First Minister said on Monday focus must turn to the UK Government’s budget on Thursday to ensure that, as the Scottish Government increases the benefit, “there’s no reason at all why the UK Government can’t and shouldn’t follow suit”.
Mr Birt continued: “A country as wealthy as the UK can do much, much better and the Scottish Government’s action shows the UK Chancellor that prioritising those on low incomes is possible.
“It also shows the other devolved administrations that constraints on powers and financial flexibility are no barrier to compassionate and significant action to support families.”
The UK Government noted its expansion of free school meals in England and other support available via Universal Credit.
A UK Government spokeswoman said: “Our priority will always be to support the most vulnerable and we recognise that people are struggling with rising prices, which is why we are protecting millions of those most in need with at least £1,200 of direct payments.
“In addition, vulnerable families are being supported by the Government’s household support fund, which was boosted by £500 million, to help pay for essentials and latest figures show that there were 200,000 fewer children in absolute poverty after housing costs compared to 2019/20.
“The UK Government has also provided an extra £123 million for the Scottish Government to help vulnerable families at their discretion and this is in addition to the significant welfare powers they already have.”
John Dickie, director of Child Poverty Action Group (CPAG) in Scotland also made a similar plea, telling politicians at Westminster: “If the Scottish government can make this kind of serious investment in protecting our children from poverty then so too can the UK Government.
“The autumn statement is the Chancellor’s opportunity to not only ensure UK benefits rise in line with inflation but to reverse the cuts made since 2010 – starting with a £20 a week uplift to child benefit.”