"My grandfather was an extraordinary man and part of an extraordinary generation," William said in his statement.
"Catherine and I will continue to do what he would have wanted and will support the queen in the years ahead.
"I will miss my Grandpa, but I know he would want us to get on with the job."
Prince William added that Philip was a constant presence in his adult life through times good and bad.
"I will always be grateful that my wife had so many years to get to know my grandfather and for the kindness, he showed her," he said. "I will never take for granted the special memories my children will always have of their great-grandpa coming to collect them in his carriage and seeing for themselves his infectious sense of adventure as well as his mischievous sense of humor."
Prince Harry, William's younger brother, remembered Philip as a "decorated serviceman, a prince and a duke."
"But to me, like many of you who have lost a loved one or grandparent over the pain of this past year, he was my grandpa: master of the barbecue, legend of banter, and cheeky right 'til the end," he said in a statement.
Harry credits Prince Philip for being "a rock" for the queen during their 73 years of marriage.
"Meghan, Archie, and I (as well as your future great-granddaughter) will always hold a special place for you in our hearts," he said.
"And while I could go on, I know that right now he would say to all of us, beer in hand, 'Oh do get on with it!'"
Prince Philip died on Friday at the age of 99.