"I remember learning how to cook, how to paint, what to read. I remember laughing at your jokes and asking about your spectacular life and service in the Navy."
Referring to the Duke's habit of hosting family barbecues at Balmoral during the summer holidays, she said: "I remember incinerating the sausages and you swooping in to save the day. I remember your hands and your laugh and your favourite beer.
"I will remember you in your children, your grandchildren and great-grandchildren.
"Thank you for your dedication and love for us all and especially Granny, who we will look after for you."
She ended the tribute, "With all my love, Eugenie."
The gallery director, who married tequila brand ambassador Jack Brooksbank at Windsor Castle in October 2018, named their first child August Philip Hawke Brooksbank after the Duke following his birth on February 9.
Prince Philip did not have the chance to meet the newborn before he passed away on Friday, aged 99.