Morgan was born in Wimbledon, London. His mother, Inga (née Bojcek), was a Catholic Pole who fled the Soviets, and his father, Arthur Morgenthau, was a German Jew who fled the Nazis, arriving in London in 1933. His father died when Morgan was nine years old. Morgan attended boarding school at Downside School, Somerset, and gained a degree in Fine Art from the University of Leeds.

Morgan wrote television scripts during the 1990s, including an episode of Rik Mayall Presents... and the Comedy Premiere The Chest. He wrote the screenplay to the romantic comedy Martha, Meet Frank, Daniel and Laurence (1998) and had some success with the TV series The Jury (2002). In 2003, Morgan broke through with The Deal, a television drama about the power-sharing deal between Tony Blair and Gordon Brown that was struck in the Granita restaurant in London in 1994.

Morgan married Anna Carolina Schwarzenberg, daughter of Czech Politician and nobleman Karel Schwarzenberg, civilly in London on 28 July 1997 and religiously at Murau on 6 September 1997. They have five children. They lived in Battersea, South London until the winter of 2006, when the family moved to Vienna. They separated in 2014 and he resides in London. Morgan has been in a relationship with Actress Gillian Anderson since 2016.

Also in 2006, Morgan's first play, Frost/Nixon, was staged at the Donmar Warehouse theatre in London. Starring Michael Sheen as David Frost and Frank Langella as Richard Nixon, the play concerns the series of televised interviews that the disgraced former President granted Frost in 1977. These ended with his tacit admission of guilt regarding his role in the Watergate scandal. The play was directed by Michael Grandage and opened to enthusiastic reviews. In May 2007, the 50th San Francisco International Film Festival honoured Morgan with the year's Kanbar Award for Excellence in Screenwriting.

In 2008 the film Frost/Nixon, with Sheen and Langella playing the parts they had on stage, opened in the UK and the US to positive reviews. Morgan was nominated for the Academy Award for Best Adapted Screenplay for his work on the film. In June 2009, it was announced that he would be the co-writer of Skyfall, the 23rd James Bond film. The Daily Telegraph's Mandrake diary reported in April 2010 that Morgan had quit the production after Sam Mendes was hired to direct, and that Patrick Marber would replace him. MGM dismissed the Telegraph's claims as speculation.

In July 2009, filming began on The Special Relationship, the third film of Morgan's "Blair trilogy". The film focuses on Blair's (again played by Michael Sheen) relationship with US President Bill Clinton between 1997 and 2000. Morgan was originally scheduled to direct the film (making his directorial debut) but pulled out a month before filming began. He was replaced by Richard Loncraine. Morgan was nominated for two Primetime Emmy Awards for his work on the film. In 2008 Morgan adapted John le Carré's Tinker, Tailor, Soldier, Spy into a screenplay for Working Title Films.

In 2010, Morgan started writing the screenplay for an upcoming biopic about rock musician Freddie Mercury, the lead vocalist, and Songwriter, for the rock band Queen.

Morgan was appointed Commander of the Order of the British Empire (CBE) in the 2016 New Year Honours for services to drama. In February 2017, he was awarded a British Film Institute Fellowship (BFI).