The cutting, which is thought to have been made in 1864, was inserted into one of the fairy tale author's first editions.
Andersen often made elaborate paper cuttings to give to friends and families.
The cutting, which measures 42cm by 34cm, was made to raise money for the victims of the Danish-Prussian war, and includes a handwritten verse written to the buyer.
The Danish handwritten verse reads: "This paper cutting is somewhat dear, The price is half a rigsdaler (the then-Danish currency), But it is a whole cut fairy tale, And your kind heart will have to pay."
It was eventually mounted on a blue cloth by the buyer, A Bruun, who pasted it into the first edition of Andersen's fairy tale book Eventyr.
Unsurprised
Auctioneers Bruun Rasmussen Auctioneers of Copenhagen had estimated the cutting, together with the first edition, would reach 150,000 kroner (£12,867).
Bids were being taken from across the world including the US and Japan.
"We are happy that we got it," said Torben Groengaard Jeppesen of the Hans Christian Andersen Museum.
"We are not surprised at all that the bids went that high up."
Born in 1805, among Andersen's most famous works are The Ugly Duckling, The Red Shoes and The Little Match Girl.
Mr Jeppesen said the cutting would go on display in the Hans Christian Andersen Museum which is due to open in 2005.