The black and white image by Chris Allerton, from Archie’s christening, shows the infant royal in his baptism robes.
Harry is seen holding his son in his arms, gazing proudly down at him, as Charles stands by his side looking at his youngest grandchild.
The Instagram message reads: "Happy birthday to His Royal Highness The Prince of Wales - Sir, Pa, Grandpa!" followed by a birthday cake emoji.
The Cambridges also sent Charles a birthday message on their official social media accounts, posting a touching image of him with Prince Louis.
The heir to the throne is seen holding Louis’ hands as the royal youngster is in Kate’s arms.
The photo was part of a series taken to mark Charles’s milestone 70th last year.
The message read: "Wishing a very happy birthday to The Prince of Wales!".
It was also accompanied by a photo of Charles and William, and another of Charles, William and Harry, in military dress, laughing together.
The prince is away on a tour of India in Mumbai, where he received a birthday present of bonsai fairy gardens from a group of Indian schoolchildren.
The Royal Family’s Twitter account posted: "Wishing The Prince of Wales a very happy birthday!".
It included a photo of a young Duke of Edinburgh holding his baby son Charles in the air.
The Duke of York’s Twitter feed shared a photo from 1960 of Charles holding baby Prince Andrew.