Burnett was born in Youngstown, Ohio, in 1838. He was the son of Henry Burnett (1801–1876), an abolitionist, and Nancy (née Jones) Burnett (d. 1880). His paternal grandfather, Samuel Burnett, a native of Morristown, New Jersey, was well educated and at one time was prominent and influential. He had a considerable fortune, but lost most of his property during the Revolutionary War, along with Robert Morris, and emigrated west to Ohio around 1798. He was also a descendant of William Burnett, a colonial governor of New York and New Jersey.