Ms White, who is currently second permanent secretary at the Treasury responsible for managing the UK's public finances, will take up her new post in late March.
She will replace Ed Richards, who will step down at the end of December, after being at the helm for eight years.
Her annual salary will be Â£275,000 a year, Ofcom said in a statement.
Ms White has spent 25 years working in the public sector and government, including spells in the British embassy in Washington DC, at the World Bank and as director general in the Department for International Development.
She is married to Robert Chote, the chairman of the Office for Budget Responsibility (OBR), which provides independent economic forecasts to the government.
"The Ofcom board is confident that Sharon will provide the leadership and vision to ensure Ofcom continues to promote a thriving communications sector in the UK that operates in the public interest," the regulator's chairman Patricia Hodgson said.
The announcement comes just a day after telecoms giant BT announced it was in talks to buy mobile operator EE, a deal expected to be scrutinised by Ofcom.
Ofcom, the UK's telecoms regulator, has named senior Treasury official Sharon White as its new chief executive.