The venue in Shaftesbury Avenue was packed for a performance of The Curious Incident Of The Dog in the Night-Time, BBC reported.
Eyewitnesses heard "a crackling" noise before the collapse at about 20:15 GMT. Theatre-goers left covered in debris.
The Apollo's owner described it as a "shocking and upsetting" incident and an investigation was underway.
A total of 51 people were taken to hospital but there were no life-threatening injuries.