The 16-year-old found herself in the bottom two alongside Danielle Hope after they both received the fewest public votes.
The two hopefuls then had to perform a rendition of 'Maybe This Time' from Cabaret to convince Andrew Lloyd-Webber to keep them in the competition.
On hearing the announcement, Rayner said: "Just to say thanks so much. I've learnt so much through this whole experience, it's been a long one but a great one. Good luck to all the girls still left in."
Lloyd-Webber said of his decision: "What I think you're a fantastic singer, I think you need to become at ease with your own body. You are the youngest, and you have a great future."
Over The Rainbow continues next Saturday at 7.15pm on BBC One.