Moore, who also shares Rumer, 32, and Tallulah, 27, with ex-husband Bruce Willis, said she felt 'honoured' to be her mother.
"You were already carving out your unique path when you decided to arrive 3.5 weeks early 30 years ago. You continue to do so with every courageous, thoughtful, and meaningful move you make in this life. I learn from you everyday. I am inspired by you. I really like you," she wrote.
"Excited to be sharing this life with you. I love you madly! I can't wait for the adventures of the next monumental 30 years - starting with the magic of today, your birthday! Happy 30th birthday, Scout LaRue. Loving you to the moon."
While the sentiment was sweet, Moore kept the post lighthearted, sharing a cheeky photo of the pair rocking swimsuits. While Scout sits in a chair, the Ghost actress shows her playful side, flashing her bum next to her bashful looking daughter.
Moore also shared a throwback photo of herself cradling Scout as a baby, as well as a snap of the birthday girl as a newborn to her Instagram stories.
The post drew plenty of birthday wishes from Hollywood, including actors Viola Davis and Debi Mazar.
Scout, who celebrated her birthday with a gathering of friends to test out an inflatable slide, added her own birthday post with a nod to her parents.
Posting a photo of herself standing between Moore and Willis, she captioned the snap: "30 years ago today the V* rare 2 of 3 DemiXBruce collab dropped."
A later post showed Scout posing in a bikini above a bridge, declaring: 'Best year yet' and another where she said: 'Grateful, hydrated, and full of love'.