It was written to author Betty Shew by the 21-year-old princess in 1947, months before her marriage.
The two-page note describes how the couple met, were chased by a photographer in Prince Philip's sports car and danced at London nightclubs.
It was given a pre-auction estimate of £800 to £1,200 but sold for £14,400.
Describing it as a "fantastic result", Richard Edmonds of the Chippenham Auction Rooms in Wiltshire said: "It was quite an honour to be able to sell such an important document, particularly as the country is celebrating the Queen's 90th birthday.
"The bidding was both online and over eight phone lines. It was pretty fast and furious. At times our internet connection looked like it was lighting up."
The identity of the buyer has not been disclosed.
The then Princess Elizabeth agreed to share the details of her relationship with Philip, for a royal wedding souvenir book being written by Mrs Shew.
In the letter, written in ink on white paper adorned with the royal crest, the princess recalls how she met Prince Philip at the age of 13, describes his love of fast cars and how the couple danced at nightclubs Ciro's and Quaglino's in London.
The future Queen also writes about her wedding ring, which she says will be made of Welsh gold.
She and Prince Philip married in November 1947 at Westminster Abbey, the same location as her coronation in June 1953.
A letter written by the Queen revealing how she and Prince Philip first fell in love attracted "furious" bidding as it smashed pre-auction estimates.