The Department for Education will fund the scheme, which builds on a commitment announced by the Treasury in March to offer free sanitary products to all female secondary school pupils from September.
Almost half (49 per cent) of students aged 14 to 21 have missed an entire day of school because of their period, according to a survey of 1,000 girls conducted by charity Plan International UK in 2017.
It also found 10 per cent of girls were unable to afford sanitary wear, and 12 per cent had to improvise sanitary wear due to affordability issues.
In his Spring statement, Chancellor Philip Hammond announced the scheme would apply to secondary school pupils, saying the move was "in response to rising concern by head teachers that some girls are missing school attendance due to inability to afford sanitary products".
Children and families minister Nadhim Zahawi said the government was "determined to ensure that no-one should be held back from reaching their potential - and wants everyone to lead active, healthy, happy lives".
"After speaking to parents, teachers and pupils, we are now extending this to more than 20,000 primary schools so that every young person in all our schools and colleges gets the support that they need," he said.
The DfE did not say how much the scheme would cost, but is currently consulting with the public and private sector on how to deliver the promise in a "cost-effective manner".
The Scottish government announced in August 2018 it would provide free sanitary products to all school, college and university students in the country at a cost of £5.2m.
And earlier this month, the Welsh government announced a £2.3m scheme offering the same support to primary- and secondary-aged girls.
Founder of the Free Periods campaign Amika George said the move would allow every child in compulsory education to attend school "without the anxiety or stress of worrying where their next pad or tampon will come from".
"This commitment will ensure that all children can fully participate in lessons and focus in class, and their period will never hold them back," she said.
"Period poverty should never be a barrier to education."
Campaigner Amika George is behind the #FreePeriods campaign
Girlguiding UK also campaigns against period poverty. Its 2018 Girls Attitudes Survey found that among girls aged 11 to 21, 30 per cent had missed education because of their period.
A 19-year-old member of Girlguiding's panel of advocates, Isla, said free menstrual products would "help break the taboo of periods from a young age".
The DfE intends for the policy to build on new relationships, sex and health curriculum which it will introduce in schools from September, and become compulsory in 2020.
Lucy Russell, head of girls' rights and youth at Plan International UK, said: "The government's announcement that more than 20,000 primary schools across the country will receive free period products is another major breakthrough for girls affected by period poverty across the UK."
Russell said that further moves should include cutting the high cost of products, tackling poor education and the stigma and shame surrounding periods.
"Free products won't solve things if girls are too embarrassed to talk about their periods or don't understand how their bodies work," she added.
"We urgently need education and training for girls, schools and parents to help tackle the stigma around periods".