The Treasury-backed provider, which has 25 million customers, said the changes, which follow last week's rise in the Bank of England base rate from 0.25% to 0.5%, will take effect from December.
In line with the 0.25 percentage point increase in the base rate, interest rates paid on Direct Isa, Direct Saver, Income Bonds, Investment Account and Junior Isa will also increase by 0.25 percentage points.
From December 1, the Direct Isa rate will increase from 0.75% to 1%, the Direct Saver will rise from 0.7% to 0.95% and the Income Bonds rate will go up from 0.75% to 1%.
The Investment Account rate will rise from 0.45% to 0.7% and the Junior Isa rate will increase from 2% to 2.25%.
The prize fund rate on Premium Bonds will also increase by 0.25 percentage points to 1.40% and the odds will improve from 30,000 to one to 24,500 to one.
The number of prizes paid out each month will increase from 2.3 million to around 2.9 million - the highest number of prizes in any monthly Premium Bonds prize draw to date, NS&I said.
The changes will come into effect from the December 2017 draw.
The shake-up means that in December there will be an estimated four £100,000 prizes, up from three in November, and around nine £50,000 prizes, up from five.
There will be an estimated 18 prizes of £25,000 in December, up from 12 in November, around 42 £10,000 prizes, up from 28, and around 87 £5,000 prizes, up from 57.
The number of £1 million prizes will remain the same, at two, according to NS&I's estimates.
NS&I chief executive Ian Ackerley said: "For our 25 million customers, including around 21 million Premium Bonds customers, these changes will present a welcome boost.
"NS&I will be giving out the largest number of Premium Bond prizes every month - an estimated 2.9 million - and all money invested is 100% secure, as NS&I is backed by HM Treasury."
NS&I has a duty to balance the interests of taxpayers and the stability of the broader financial services sector, while also meeting the needs of savers.
Its announcement comes after a wave of banks and building societies said they also plan to tweak savings rates upwards following the base rate increase.
Sarah Coles, a personal finance analyst at Hargreaves Lansdown, said: "Savers will be pleased so see so many providers are passing on the rate rise in full, for at least some of their accounts, and NS&I savers will be delighted to see it join the fray."
She continued: "Even after all of these announcements, savers should still check the rate they are receiving against the most competitive in the market. Even if your rate has risen, you may be able to beat it by shopping around.
"Over the past few days we have seen the release of some highly competitive fixed-rate deals - and the rates available on the most competitive one-year bonds are edging ever-closer to 2%."