Elwes (pronounced "El-wez") attended Harrow School in England, and Sarah Lawrence College in Yonkers, New York, where he studied, among other acting-related subjects, German Expressionist Film.

Elwes descends from a staunchly Roman Catholic recusant family on his father's side (originally known as the "Cary-Elwes" family, but now many branches of the family are now simply known as "Elwes") which includes noted British monks and bishops, such as Abbott Columba Cary-Elwes, Archbishop Dudley Cary-Elwes and Father Luke Cary-Elwes.

His artist father, Dominick Elwes, had moved in the circles of high society (and been friends with Lord Lucan and Lady Annabel Goldsmith to whom he was also related) before he took his own life in 1975.

His mother, interior designer Tessa Kennedy, also a Catholic, eloped with his father in 1957 as a teenager to great scandal. She is of Croatian and Anglo-Irish extraction. She has been married to Jewish-American Hollywood executive, Elliott Kastner, for many years.

Elwes became engaged to the photographer Lisa Marie Kurbikoff in 1997 in Paris. The wedding took place in 2001 in California.

Elwes has 2 brothers, Cassian and Damian, who are also involved in the entertainment industry.

Elwes filed a lawsuit in August 2005 against his management firm and the producers of Saw. He alleged that he was promised "a minimum of one percent of the producers' net profits" and did not receive the full amount as promised .

Elwes had a recurring role in the final season of The X-Files as FBI Deputy Director Brad Follmer.