The pair, both 28, wrote on Instagram that the child had been born on Saturday.
They shared black and white images of the child’s head and feet, but did not reveal a name or gender.
Edwards had announced she was pregnant in May, writing that she was “so happy to be on this wild journey with my soulmate.”
She received congratulations from many of her 12.7 million Instagram followers, including “Love Island” presenter Laura Whitmore and singer Stacey Solomon.
Meanwhile, Oxlade-Chamberlain was congratulated by England teammates Mason Mount, Trent Alexander-Arnold and Jordan Henderson.
Little Mix shot to fame in 2011 after they won the UK’s “The X Factor” TV talent show.
They have gone on to become one of Britain’s biggest girl groups, with five number one singles in the UK, and one number one album.
The remaining members of the group, Edwards, Leigh-Anne Pinnock and Jade Thirlwall, said they intended to continue performing and touring after Nelson’s departure.