"I've never done anything like this before and in hindsight I have realised it was a silly thing to do."
The spokesman said police were called because of her "misguided actions" but no further action would be taken against her.
Security were quick to react, escorting her off the stage, but only after one of her missiles hit Cowell on his jacket.
Cowell reacted calmly, joking to the audience he thought the woman was throwing sandwiches and it was part of the performance.
He said: "I sent a tweet out saying I don't like eggs, and I really don't like eggs now."
After the Welsh brothers' performance, presenters Ant and Dec said: "We'd just like to point out that was not part of the performance. Well done for carrying on."
Fellow judge Alesha Dixon apologised to the group for the interruption and Amanda Holden was less polite, calling the woman a "stupid cow".