The 30-year-old has been cast as the psychopathic serial killer Patrick Bateman in the adaptation of Bret Easton Ellis's 1991 thriller.
The production will run at London's Almeida Theatre from 3 December to 25 January 2014.
The show almost completely sold out within hours of Smith's involvement being revealed.
Set in the 1980s, American Psycho tells the story a 26-year-old Manhattan investment banker whose sanity slowly disintegrates, turning him into a brutal murderer.
It was previously adapted for the big screen in 2000 with Christian Bale in the lead role.
The show, which was announced minus cast details in January, will be directed by the Almeida's new artistic director, Rupert Goold, who developed the musical with his former company, Headlong.
Tony Award-winning composer Duncan Sheik has collaborated with Spider-Man: Turn Off the Dark scriptwriter Roberto Aguirre-Sacasa to adapt the book for the stage.
Smith said he was "excited" to be part of the production.
"The opportunity to work with Rupert Goold, a director of great vision and flair, combined with the brilliant story and music, seemed like such a cool challenge," he said.
Goold added: "Matt Smith is a wonderful stage performer and the perfect actor not only to lead our exceptional cast but also my first show as artistic director here at the Almeida."
Smith has just finished filming his final scenes for the Christmas special of Doctor Who, which will see his character regenerate into the 12th Doctor, played by Peter Capaldi.
Doctor Who actor Matt Smith is to star in a musical version of the controversial novel American Psycho.