Essenhigh was born in Newcastle upon Tyne and educated at St. Cuthbert's School. He joined the Royal Navy in 1963, and was promoted to sub-lieutenant on 4 June 1965 and to lieutenant on 1 May 1967. He qualified as a principal warfare officer in 1972, specialising in navigation. He served as a navigating officer on the frigate HMS "Juno" and then the destroyer HMS "Antrim" before joining the staff of the Flag Officer Sea Training. Promoted to lieutenant commander on 1 May 1975, he was posted to the destroyer HMS "Glasgow" in 1978.