Smits will repeat the role he is currently playing at New Jersey's McCarter Theatre, where Anna continues until Oct. 19.
The box office for the Broadway production opened on Sept. 22.
The show will officially open on Nov. 16.
Playwright Cruz has set his play in a Cuban-American cigar factory in 1929. The factory employs "lectors" to entertain and educate the factory workers, and a new lector who reads aloud from Anna Karenina becomes "a catalyst in the lives of his avid listeners, for whom Tolstoy, the tropics, and the American dream prove a volatile combination."