He was born in Liverpool, Lancashire, and studied at the Royal College of Music. His father was violinist and conductor Eugène Goossens, his brother the conductor and composer Eugene Aynsley Goossens and his sisters the harpists Marie and Sidonie Goossens.
During the early and middle parts of the 20th century, he was considered among the premier oboists in the world. He joined the Queen's Hall Orchestra (conducted by Henry Wood) at the age of 15 and was later (1932) engaged by Sir Thomas Beecham for the newly founded London Philharmonic Orchestra, but he also enjoyed a rich solo and chamber-music career. He became famous for a uniquely pleasing sound no other oboist could match. Oboists of the past had tended to be divided between the French school (elegant but thin and reedy in tone) and the German (full and rounded but rather clumsy, with little or no vibrato), but Goossens brought together the best qualities of both styles.
Goossens commissioned a number of works for the oboe from such distinguished composers as Sir Edward Elgar, Ralph Vaughan Williams and Rutland Boughton and collaborated extensively with other prominent soloists such as Yehudi Menuhin. Amongst his many pupils were the oboists Evelyn Barbirolli, Joy Boughton, daughter of Rutland Boughton and Peter Graeme, oboist of the Melos Ensemble.
He was appointed a Commander of the Order of the British Empire (CBE) in 1950 and made a Fellow of the Royal College of Music in 1962.