"Welcome to the world Sophia, our beautiful baby girl, born yesterday 6lb 9oz. Proudest day of our lives, never felt love like this," the 29-year-old daughter of Formula 1 CEO Bernie Ecclestone captioned the first image.
"Three musketeers forever. Love my little family x," she wrote along another photo that included her husband.
Ecclestone and former Pacific Continental senior broker Rutland became engaged in February 2013 having known each other for one month. The couple tied the knot on the French Riviera in June 2013.