Abse was born in Cardiff, Wales to a Jewish family. He is the younger brother of politician and reformer Leo Abse and the eminent psychoanalyst, Wilfred Abse. Abse first studied medicine at the University of Wales, and then at Westminster Hospital and King's College London.

Although best known as a poet, Abse is also prominent in the medical field, and was a specialist at a chest clinic for over thirty years. He has received numerous literary awards and fellowships for his writing. In 1989, he received an honorary doctorate from the University of Wales.He is also a talented musician, having learned the flute at the age of six.His first poetic volume, After Every Green Thing, was published in 1949. His autobiographic work, Ash on a Young Man's Sleeve, was published in 1954. He won the Welsh Arts Council Award in both 1971 and 1987, and the Cholmondeley Award in 1985. He has been a Fellow of The Royal Society of Literature since 1983.

Abse has lived for several decades in the northwest area of London, mainly near Hampstead where he has considerable ties. For several years he wrote a column for the "Ham & High" (Hampstead and Highgate Express) local newspaper. The articles were subsequently published in book form.

In 2005 his wife Joan Abse was killed in a car accident, while Abse suffered a broken rib. His poetry collection, Running Late, was published in 2006, and The Presence, a memoir of the year after his wife died, was published in 2007; it won the 2008 Wales Book of the Year award Dannie Abse. The book was later dramatised for BBC Radio 4.

In 2009 Abse brought out a volume of collected poetry. In the same year, he received the Wilfred Owen Poetry Award.