Viola player Natalie Holt was bundled off stage by security guards after running to the front while finalists Richard and Adam sang an operatic version of The Impossible Dream.
The 30-year-old musician's band Raven Quartet was part of the orchestra accompanying the Welsh opera brothers during their performance on Saturday night.
She was reportedly a contestant on the show last year and was not put through during auditions.
In a statement issued by a BGT spokesman afterwards, Ms Holt said: "I want to apologise to Richard and Adam for overshadowing their performance.
"I've never done anything like this before and in hindsight I have realised it was a silly thing to do."
BGT said police were called after Ms Holt's "misguided actions", but added that it would not be taking any further action against her.
Cowell, who had subsequently removed his jacket, attempted to brush off the episode with a joke.
He said he had previously tweeted that he did not like eggs and added: "I really don't like eggs now. I have no idea what that was about."
However, fellow judge Amanda Holden appeared furious, and reacted: "The only person with egg on their face is the stupid cow who threw the eggs."
Comic David Walliams, another judge tweeted: "Now you all know it is live...! One of the string quartet apparently. Well she has had her 15 seconds of fame...!"
Twitter was also awash with tweets from viewers reacting to the stunt.
One said: "Crazy egg thrower Natalie Holt highlighted a very serious security lapse - done them a favour could have been guns not eggs."
Another tweeted: "Still in shock about the egg incident. She look possessed!!!"
Richard and Adam, who did not stop singing during the egg-throwing, finished third overall in the competition - won by Hungarian shadow dance troupe Attraction.
The group, whose presence on a show supposedly highlighting native performers had led to criticism, won the public vote, picking up £250,000 and a slot performing at the Royal Variety Performance.
Teenage comic Jack Carroll, 14, from Huddersfield, who has cerebral palsy, was named runner-up.