After appearing in the British television series The Knock (1996) and Fish (2000) during his teenage years, he had his first starring role in the American slasher film Cry Wolf (2005).

He subsequently had supporting roles in the thriller Donkey Punch (2008), the historical drama Valkyrie (2008), and another slasher film Sorority Row (2009).

Morris has also appeared in American television as a series regular on Pretty Little Liars and Hand of God, and had recurring roles on Once Upon a Time and New Girl.

In this period more opportunities arose, and after appearing in several television shows, films and commercials, he went on to spend three seasons with the Royal Shakespeare Company, working with directors such as Steven Pimlott, Tim Albery and Elijah Moshinsky, and alongside actors like Simon Russell Beale, Samuel West and Roger Allam.

Morris was part of the main cast of the 2010 ABC documentary-style dramedy series My Generation, playing "The Rich Kid" Anders Holt.

In May 2016, Morris landed a major role in the film Mark Felt: The Man Who Brought Down the White House playing Bob Woodward, the Washington Post reporter who teamed with Carl Bernstein to expose the Watergate dealings.