Paul posted an adorable black and white photo of the baby's foot. "My heart," he wrote.
"There are no words that will ever be able to explain what just happened to my heart," Lauren wrote on her page. She posted two images - one showing her daughter's face and another showing mom and baby in the hospital bed.
The couple did not share their baby's name, but he told PEOPLE in December that they did have a few names in mind.
"We just hang out in our nursery a lot," he told PEOPLE. "It's our favorite room of the house, and yeah, everything's ready. We put in our car seat today, which was very exciting.
Paul was on Jimmy Kimmel Live! last month, and told the host he is "over the moon excited" about becoming a parent.
"I love babies," the 38-year-old told Kimmel. "I always have loved babies."
Paul said they hired a "doula," a trained professional who helps mothers through the birth and postpartum experience, according to DONA International.
"Do you know what a doula is?" Paul asked Kimmel. "I had no idea what a doula was, but she's teaching us some stuff."
Paul said their doula was an incredible help during the process.
"She said, 'First thing you need to know is never say the word relax and don't ever say breathe.' She said that my wife would most likely hit me if I do. So, I'm like, 'OK, that's good to know,'" Paul said.
He continued, "She's just telling me what not to do. She goes, 'Just prepare yourself. I've helped deliver over 600 babies and 100 percent the mother of the baby who's delivering absolutely hates her partner."
Paul is best known for his Emmy-winning performance as Jesse Pinkman on AMC's Breaking Bad. He now stars on Hulu's The Path.
Photo credit: Joe Seer : Shutterstock.com