The three-quarter length drawing in pen and brown ink - Study of a Mourning Woman - was expected to raise up to £8m when it was sold by Sotheby's auction house in London.
The sketch, which lay unrecognised for more than 250 years, is said to date from between 1494 and 1504 and is regarded as a major work by one of the greatest artistic geniuses in history.
Excluding the Michelangelo drawings in the Royal collection, only three other important drawings by the artist are known to be still in private hands.
Speaking about the drawing, a Sotheby's spokesman said: "It adds greatly to our knowledge of the earliest works of this great artist, and is also an extremely beautiful and powerful image."
Scrapbook
It was a Sotheby's expert, Julien Stock, who chanced upon the work while flicking through a scrapbook of Old Master drawings kept in the library of the stately home for an insurance evaluation.
"Mr Stock recognised it as belonging to a small group of large-scale figure studies by Michelangelo, which the artist is thought to have made early in his career, between about 1495 and 1505," said the spokesman.
"The drawing bears the collector's mark of the 18th century artist and connoisseur Jonathan Richardson Senior.
"Although there is no record of exactly when the drawing entered the
collection at Castle Howard, it was most probably purchased at the 1747 London auction of Richardson's celebrated drawings collection, where Henry Howard, 4th Earl of Carlisle, is known to have been an active buyer."
At the time Sotheby's described the find as "the most significant Michelangelo find in living memory; like finding part of the Holy Grail".
The cost of insuring such a valuable work made it impossible to keep the drawing at Castle Howard, which was the setting for the TV series Brideshead Revisited.
In March the National Galleries of Scotland launched an appeal to raise money to prevent the work from being sold to a private collector or abroad.
Michelangelo was a leading figure in the High Renaissanace and painted the ceiling of the Sistine Chapel in Rome as well as sculpting such works as David.