Born weighing just over 3.6kg, the baby is the ninth great-grandchild for the 94-year-old queen.
"The Queen, the Duke of Edinburgh, the Duke of York, Sarah, Duchess of York, and Mr and Mrs George Brooksbank have been informed and are delighted with the news," the palace said.
Eugenie, who married Brooksbank at Windsor Castle in 2018, announced the news on her Instagram account with a picture of the new parents holding the newborn's hand.