In a Yorkshire Post article, the prime minister promised to "help the great cities and towns of the North pool their strengths and take on the world".
This includes backing a bid to bring cycling's world championships to Yorkshire.
She also said she would make the Midlands an "engine for growth".
The Northern Powerhouse is the name attached to the government's bid to create a rival economy to London and the South East in the north of England.
It involves devolving new powers from Whitehall, improving transport links and creating new regional mayors to act as figureheads for their regions.
Mrs May's language was similar to that used by Mr Osborne when he set out his vision to create "a collection of northern cities sufficiently close to each other that combined they can take on the world".
On taking over as prime minister, she appointed Andrew Percy, the MP for the Yorkshire and Lincolnshire constituency of Brigg and Goole, as her Northern Powerhouse minister.
Writing in the Yorkshire Post, she promised investment in the county would not be "disrupted" by leaving the EU, saying the government would "underwrite" EU investment.
She also announced Â£24m government backing for a bid to bring the cycling world championships to Yorkshire.
"Our backing will include Â£15m for cycling infrastructure projects, to encourage even greater participation in the sport and continue the proud legacy that has seen our athletes excel at the Rio Games," she added.
In a separate article in the Birmingham Mail, the PM said the city had "untapped potential" and that the Midlands would be a "key part" of her government's industrial strategy.
Theresa May has vowed to press ahead with George Osborne's "Northern Powerhouse" despite having fired him as chancellor.