The former England football captain shared a photo of the young couple embracing on his profile, with the caption: "Congratulations to these two beautiful people.
"As they begin this exciting journey together we are so so happy for you guys."
Posting a photo of himself and Ms Peltz, he wrote: “Two weeks ago I asked my soulmate to marry me and she said yes. I am the luckiest man in the world.
“I promise to be the best husband and the best daddy one day. I love you baby.”
Ms Peltz, 25, shared the same picture on her own account, hinting that it had been taken by Beckham’s nine-year-old sister Harper.
She wrote: “You’ve made me the luckiest girl in the world. I can’t wait to spend the rest of my life by your side. Your love is the most precious gift.
“I love you so so much baby and thank you harper for this pic.”
Mum Victoria shared her delight at the news, posting the same image.
The former Spice Girl and fashion designer wrote: “The MOST exciting news!! We could not be happier that @brooklynbeckham and @nicolaannepeltz are getting married!
"Wishing you so much love and a lifetime of happiness. We all love you both so much"
A host of celebrities also congratulated Beckham, who turned 21 in March this year, and his bride-to-be.
Australian singer Cody Simpson, who is dating Miley Cyrus, wrote: “Congratulations my brother. So happy for you! Wishing you two a lifetime of happiness”.
Actor Jack Whitehall and singer Yungblud also both posted congratulatory messages.
Peltz has starred in TV series Bates Motel and the film The Last Airbender and in January this year she and Beckham confirmed their relationship with posts on Instagram.
They have been dating for several months following his split from British model Hana Cross last summer.