Gilpin was born 30 Oct 1733 in Cumbria, the son Captain John Bernard Gilpin, a soldier and amateur artist. His elder brother William Gilpin was a clergyman, schoolmaster, and author of several influential works on picturesque scenery.
Apprenticed to the marine painter Samuel Scott of Covent Garden, Sawrey came to specialise in painting animals, particularly horses and dogs, which he sometimes added to backgrounds by other artists, including Philip Reinagle, George Barret and J. M. W. Turner. He was patronised by Prince William Augustus, Duke of Cumberland. Gilpin was Director and President of the Society of Artists, and a member of the Royal Academy from 1796.
He died at Broughton, Northamptonshire, England in 1807.