The Duke and Duchess of York said their younger daughter and Mr Brooksbank became engaged while in Nicaragua earlier this month.
The wedding will take place at St George’s Chapel, Windsor Castle, in the autumn.
The princess met the former manager of Mayfair club Mahiki while skiing. They have dated for several years.
It will be the second royal wedding at the chapel this year – Prince Harry and Meghan Markle will marry there in May.
A Buckingham Palace spokesman said Princess Eugenie’s grandparents, the Queen and the Duke of Edinburgh, were “very pleased and wish the couple all the best”.