The baby girl and baby boy, whose names have yet to be announced, join Luyendyk and Burnham's 2-year-old daughter, Alessi.
In the first of the pics shared by Luyendyk, Burnham, in a hospital gown and hairnet, smiles as one baby rests on her chest and the other is held by her head.
Another shot shows one of the newborn's holding their mom's finger, while the third and fourth pics are solo shots of the newest arrivals.
"Never been prouder of my wife than in this moment. She is the strongest woman I know and I’m lucky to be on this incredible journey with her," Luyendyk gushed in the caption. "We are proud parents of a healthy boy and girl!"
"We did it babe!" Burnham commented. "I love you so much."
Burnham posted pics as well, sharing one of the same pics her husband did, as well as an adorable new shot of the babies holding hands as they lay side-by-side. She captioned her post with the twins' birthday.
Comments poured in from Bachelor Nation, with Jade Roper calling Burnham "incredible," and her husband, Tanner Tolbert, writing, "Congrats!"
"Oh my gosh they are perfect," new mom Lauren Bushnell marveled. "Congrats mama!"
"They are beautiful!! Congratulations!" Tenley Molzahn agreed. "Sending you so much love and prayers as you all transition to this new exciting adventure from family of 3 to 5!!"
Kristina Schulman chimed in too, writing, "Congratulations to you and Arie. Beautiful babies, beautiful growing family."
Raven Gates, Hannah Ann Sluss, Vanessa Grimaldi and more also expressed their well wishes.