The sepia-toned photo of Archie staring at the camera and gripping his dad's left hand was shared on the SussexRoyal Instagram account on Sunday.
Prince Harry's wedding ring is also visible in the picture.
The post was captioned: "Happy Father's Day! And wishing a very special first Father's Day to The Duke of Sussex."
Meghan and Harry also shared an image to mark American Mother's Day in May when Archie was less than a week old.
The post of the baby's feet in front of forget-me-not flowers was liked more than two million times on the social media site.
It was captioned: "Paying tribute to all mothers today, past, present, mothers-to-be, and those lost but forever remembered.
"We honour and celebrate each and every one of you."
The Prince of Wales and the Duke of Cambridge also shared Father's Day posts on Sunday.
In a message on the Clarence House Twitter page, Charles shared good wishes with "Dads everywhere" alongside an image of him with his two sons.
William marked the occasion by sharing two photos - one with him walking beside Charles and a more recent picture with Prince Louis playing on a rope swing in the Duchess of Cambridge's Chelsea Flower Show garden.
Harry and his American former actress wife have been enjoying life as a family of three after welcoming their first born at the start of May.
Smitten Harry introduced Archie to the world alongside proud mother Meghan when the newest addition to the royal family was just two days old.
Announcing his arrival on May 6, the duke said: "This little thing is absolutely to-die-for, so I'm just over the moon."
The Sussexes are raising seventh in line to the throne Archie in the sanctuary of Frogmore Cottage - their new bolthole in Windsor Home Park in Berkshire, close to the Queen's residence Windsor Castle.
The Grade II listed 10-bedroom property is set close to the winding lakes, wooded mounds, glades, walks and bridges of the gardens at Frogmore.
Hands-on-father Harry has been relishing his new role as a parent after years of watching others bring up their families.
The duke said at Archie's first photocall at Windsor Castle: "It's great. Parenting is amazing.
"It's only been two-and-a-half days, three days, but we're just so thrilled to have our own little bundle of joy and be able to spend some precious times with him as he slowly, slowly starts to grow up."
Meghan said of her husband and her son: "It's magic, it's pretty amazing. I have the two best guys in the world so I'm really happy."
Harry previously spoke about his desire to start a family, saying: "I've longed for kids since I was very,