"Im so in love!!!!" Liv wrote alongside a picture which shows two of her fingers touching the little boy's bare feet.
The 37-year-old later paid tribute to Sailor's dad with a picture of Dave standing outside a shop where they were picking up supplies for their little one.
"Who's your daddy!!! @davidgardner76 beautiful father best friend Thank you for making me smile so big and laugh so hard and loving me so much," she wrote.
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Liv Tyler paid tribute to her boyfriend Dave Gardner on Instagram
"Sailor Gene Gardner @davidgardner76," she wrote, posting a sweet photo of baby Sailor clutching a blue pendant with an anchor symbol. Dave echoed his partner's post, sharing the same adorable photo with his followers.
The couple's first child together was born six weeks early, meaning both Liv and Sailor spent some time in hospital after the birth.
Despite arriving earlier than expected, Sailor is doing well and a representative for the couple told People: "He's healthy and everyone is happy".
Liv Tyler and Dave Gardner have been dating since 2014
Liv and Dave have since been spotted out and about in New York with their baby boy, and it appears they may have more happy news to share. Liv has been seen wearing a beautiful diamond ring on her engagement finger, sparking reports that her partner has popped the question.
Leftovers star Liv has been dating Dave – who is David Beckham's best friend – since the start of 2014 after being introduced by mutual friend Kate Moss. They went public with their relationship in August that year, with a source telling The Sun: "This is the real deal. Dave and Liv are madly in love."