It will explore the relationships between Charlotte, Emily and Anne and their brother Branwell, who was latterly an alcoholic and drug addict.
All three sisters managed to produce great literary works before their untimely deaths.
Wainwright said she was "thrilled".
The Bafta-winning writer, whose other credits include TV series Happy Valley, described the sisters as "fascinating, talented, ingenious Yorkshire women".
•Charlotte, Emily and Anne Bronte were 19th-Century novelists who formed one of the world's most famous literary families.
•Often left alone together in their isolated Haworth home, all three sisters began to write stories at an early age.
•Charlotte's Jane Eyre and Emily's Wuthering Heights are hailed as British classics. Anne's The Tenant of Wildfell Hall was a huge bestseller.
•Tragedy struck the family when Emily and Anne both died of tuberculosis within six months in 1848-49. It also killed their brother, Branwell.
•Charlotte continued to write and later married, but she too was killed by the disease in March 1855.
Source: BBC History
To Walk Invisible: The Bronte Sisters will be filmed in and around Yorkshire, where they lived.
Charlotte, whose works included Jane Eyre, died aged 38; while Emily, who wrote Wuthering Heights, died aged 30; and Anne, who died aged 29, wrote The Tenant Of Wildfell Hall.
They all struggled with their brother's behaviour, which became increasingly difficult after a failed love affair.
BBC One controller Charlotte Moore said: "It's an extraordinary tale of family tragedy and their passion and determination, against the odds, to have their genius recognised in a male 19th-Century world."
The programme will also explore how their self-educated father, who grew up in poor, rural Ireland, encouraged his children to become passionate about literature.
Casting has yet to be announced.
A drama about the "tragedy and passion" of the difficult lives of the Bronte family is to appear on BBC One, written and directed by Last Tango In Halifax author Sally Wainwright.