Cranston, former star of TV's "Breaking Bad" and "Malcolm in the Middle," made his Broadway debut in "All the Way." He previously starred in the stage drama when it ran last year at the American Repertory Theatre in Massachusetts.
The nearly three-hour play dramatizes Johnson's first year in office as he deals with the aftermath of the John F. Kennedy assassination and the growing civil rights movement.
Cranston is a Los Angeles native who got his early start on the stage in regional theaters around Southern California. In 2006, the actor appeared in a Geffen Playhouse production of Sam Shepard's "The God of Hell" that was directed by Jason Alexander.
Last year's winner was Tracy Letts for his performance in the 2013 revival of "Who's Afraid of Virginia Woolf?"