Fire crews and 25 ambulances are at the scene and some people have been removed on stretchers.
All those trapped by the debris have now been freed, said London Fire Brigade.
Eyewitnesses said people had left the theatre crying and coughing and that some were bleeding.
Eighty-one people are "walking wounded" according to the London Ambulance Service, with seven of them "more seriously" injured.
London Fire Brigade said "heavy rafters and plaster fell from the roof".
The Curious Incident of the Dog in the Night-Time was playing to more than 700 people when the ceiling came down around 30 to 40 minutes into the performance.
Eyewitnesses said they heard a creak prior to the collapse, which left the auditorium engulfed by dust.
Martin Bostock, who was in the theatre, told Sky News: "I was in the stalls with my family in the early stages of the show.
"I think the front part of the balcony fell down. At first we thought it was part of the show, it was very dramatic.
"We got out with cuts and bruises, I think most people did."
Simon Usborne, also watching the performance, told Sky News: "A huge crack happened ... and then my whole view of the stage was obscured by a huge cloud of dust.
"I think everything in the front row of the stalls and the dress circle would have been showered with debris."
Mr Usborne said people were being treated for head injuries.
"There are a lot of people with blood on their faces," he added.
"The Curious Incident of the Dog In The Night-Time is pretty popular with families and they have left in tears, they are pretty distraught."
Another witness, William Rolt, said: "Everyone who was in there wanted to get out as quickly as possible but you couldn't anything it was pitch black in there.
"It was complete pandemonium.
"Now people are just looking for their kids, trying to get reunited with their friends and family."
Forty walking wounded have been taken to the nearby Gielgud Theatre for treatment and a London bus also took some people to hospital.
'Hannah' - who was sitting in the balcony - told Sky News the incident happened without much warning.
She said: "The first few rows started getting up very quickly and the ceiling just came down".
Sky News' Darren McCaffrey said people had been asked to leave nearby restaurants and theatres.
Structural engineers are expected to spend most of the night assessing the safety of the Apollo.
The theatre - a Grade II listed building - was completed in 1901 and seats 755 people.
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