The newborn girl, who was delivered via surrogate, is the first biological daughter for the pair. She is a little sister to George's three adopted children Amanda, 32, Katie, 25, and 20-year-old Jett.
Their bundle of joy was welcomed to the world on 9 August, according to the Huffington Post.
George, 69, and Mellody, 44, tied the knot at the end of June against the backdrop of George's Skywalker Ranch in California.
The ceremony was described by Star Wars director Ron Howard "nothing short of profound" and he said the day was "a joy to behold".
A glittering guestlist included industry legends Steven Spielberg, Oprah Winfrey, Robert De Niro, Harrison Ford and Calista Flockhart.
Samuel L Jackson, who played Jedi Mace Windu in various Star Wars blockbusters, tweeted his congratulations to the new Mr and Mrs Lucas." Let's give a Galactic shout out to Master George Lucas & his Bride Melodie on This their WEDDING DAY!! Congrats!!!!!!," he wrote on Twitter.
The stunning bride wore a fairytale bustier gown by Peter Soronen. The intricate embellished detailing dazzled underneath the warm West Coast sunshine.
It was the second marriage for 69-year-old George and the first for 44-year-old Mellody who is chair of the board at Dreamworks Animation.
The newlyweds have been together since 2006.
They spoke candidly about their relationship in an interview last year.
"I think it works because we are extraordinarily open-minded people and we're open to what the universe brings us," Mellody told Oprah Winfrey.
"I think we didn't have preconceived ideas about what a partnership should be and so we allowed ourselves to discover something that was unexpected."