At the height of the incident 20 fire engines and 100 firefighters were tackling the blaze. Around three quarters of the building roof was damaged by fire.
The Fire Brigade was called at 1106 and the fire was brought under control at 1830. Crews from across London attended the incident.
Station Manager Tom Goodall who was at the scene of the fire said: "Our crews worked really hard in hot and taxing conditions to bring this fire under control as quickly as possible. It was a difficult job but firefighters managed to prevent the fire from spreading to the neighbouring properties. The building itself, which is under refurbishment, was evacuated quickly and there were no reports of any injuries."
There were no reports of people trapped or injured and the cause is unknown. London Ambulance Service sent paramedics and five specialist hazardous area response team units as a precaution.
The fire at the building - which was the site of the first BBC broadcast in 1922 - sent smoke across the London skyline.
Customers with Dirty Dancing tickets should contact their point of purchase to change their booking to another performance or request a refund.
No other London productions were affected.