Directed by Gordon Edelstein, the play is to begin previews Dec. 16, with opening night planned for Jan. 17, at Manhattan's American Airlines Theatre, the Roundabout Theatre Company said Tuesday.
The limited engagement is to play through March 4.
Harris is to play Miss Helen, Gugino is to play Elsa Barlow, and Dale is to play Marius Byleveld.
The playwright's "Master Harold … and the Boys" was produced at Roundabout in 2003. He directed the world premiere production of "The Road to Mecca" at the National Theatre in London and at the Promenade Theatre off-Broadway.
Fugard, who is to turn 80 next June, played the role of Marius Byleveld in both productions, which won the 1988 Drama Critics' Circle Award for Best Foreign Play.
"Set in the region of South Africa known as the Karoo, 'The Road to Mecca' tells the story of an elderly woman who has spent the years since her husband's death transforming her home into an intricate and dazzling work of art," the Roundabout said in a synopsis.
"The reclusive Miss Helen has become depressed and appears increasingly unable to care for herself. Pastor Marius Byleveld, who embodies the village's conservative values, is determined to get Miss Helen into an old-age home. Her friend Elsa, a young teacher from Cape Town who is deeply suspicious of the patriarchal traditions Byleveld represents, is just as determined that Miss Helen remain free."