Elwes was born in London. His advertising agent father Dominic Elwes and grandfather Simon Elwes were very well-known British portrait painters. His father 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 committed suicide in 1975.

His mother, interior designer and shipping heiress Tessa Georgina Kennedy, is of Croatian and Anglo-Irish descent; she eloped with his father in 1958 at age nineteen to great scandal. Elwes's mother later re-married to Hollywood executive, Elliott Kastner, from whom he has a half-sister, Milica Kastner and half-brother Dillon Kastner.

Elwes made his acting debut in Mariek Kanievska's famous film Another Country (where he played James Harcourt, a young and sentimental gay boy from an Eton-esque traditional English boarding School), with Rupert Everett and Colin Firth. His first notable role was Guilford Dudley in the highly acclaimed British epic Lady Jane after being personally chosen by distinguished Director of the Royal Shakespeare Company, Trevor Nunn.

After Elwes' success with this film, he turned down Trevor Nunn's offer to join the Royal Shakespeare Company in order to continue work in movies. His performance in Lady Jane led Director Rob Reiner to cast him in another leading role, this time as the stable boy turned hero Westley in the American comedy The Princess Bride, which required Elwes to do all of his own swordfighting and acrobatics.

Elwes' breakthrough performance in The Princess Bride gained international acclaim and was his first major attempt at comedy. Prior to this film, he had no comedic experience, having only studied and worked in drama.

Elwes continued working steadily, varying between dramatic roles, as in the Academy Award-winning Glory, and comedic roles, as in Hot Shots!. He starred as Robin Hood in Mel Brooks' Robin Hood: Men in Tights, parodying Errol Flynn and Kevin Costner in The Adventures of Robin Hood and Robin Hood: Prince of Thieves, respectively. ("Unlike other Robin Hoods," he quips in the film, taking aim directly at Costner and Flynn, "I can speak with an English accent!").

Elwes continued to appear in such popular movies as Bram Stoker's Dracula, The Crush, The Jungle Book, Twister, Liar Liar, Cradle Will Rock, The Cat's Meow and Kiss the Girls. He did not appear in another blockbuster until 2004 in the horror/thriller Saw. The same year he appeared in the film Ella Enchanted, portraying the villain rather than the hero.

In 2007, he made another "bad guy" appearance in Georgia Rule, portraying a man discovered to be molesting his step-daughter.