Fallon will chat with co-creators Steve Martin and Edie Brickell, and the cast, led by Carmen Cusack, will perform "Sun's Gonna Shine."
The Tonight Show airs on NBC at 11:35 PM ET; check local listings.
With direction by Tony Award winner Walter Bobbie and choreography by Josh Rhodes, Bright Star will officially open March 24 at the Cort Theatre.
The new musical, according to press notes, "tells a sweeping tale of love and redemption set against the rich backdrop of the American South in the 1920s and ‘40s. When literary editor Alice Murphy meets a young soldier just home from World War II, he awakens her longing for the child she once lost. Haunted by their unique connection, Alice sets out on a journey to understand her past — and what she finds has the power to transform both of their lives."