Garnett was born at Otley in Yorkshire on 25 July 1789, the eldest son of a paper manufacturer, William Garnett.
In 1834 he married Rayne, daughter of John Wreaks, esq., of Sheffield, and in 1836 was presented to the living of Chebsey, near Stafford, which he relinquished in 1838, on succeeding Henry Francis Cary, the translator of Dante, as assistant-keeper of printed books at the British Museum.