Elia was previously head chef at Delfina Restaurant (Delfina Studio Café) on the capital's South Bank.
With help from her Delfina sous chef Daniel Ferreira, who has also made the move, Elia will assume responsibility for the culinary direction of Gallery Dining Room, which opens 14 April, as well as the café bar and events business at the East End gallery.
The Gallery Dining Room will champion British food and its 40-seat interior has been designed by Project Orange.
Elia will offer a "short but sweet" seasonal menu that will change on a regular basis. A daily 2-course special lunch menu will be available for £15.
She said: "I see huge scope for the Whitechapel Gallery to position itself not only as a leading cultural destination but also as a sought after venue for dining, events and parties."
The expanded Grade II-listed building reopens to the public on the 5 April after a £13.5m investment and features new galleries dedicated to presenting collections and new commissions; a permanent gallery and research room for the Whitechapel Gallery's historic archive, and an education and research tower including study and creative studios.
Elia, who has experience at El Bulli and Arzak restaurants in Spain, will have her first book published next month entitled The Modern Vegetarian.
Maria Elia recipes:
By Chris Druce