Hawn, who co-hosted the episode with Hudson, said she asked her daughter's permission to attend the birth after eating pizza and Doritos in the delivery room during Hudson's son Bingham's birth.
"I was careful and I said, 'Would it be okay? Do you want me in the room?'" Hawn said. "So, I did ask permission. See, I'm a really fair mom, aren't I? It's like, 'Honey, so you still want me?'"
Hudson hoped to give birth to her baby girl naturally but ended up having a walking epidural.
"The doctor was amazing. But mom, once again -- she was right in there!" the star said.
"[The doctor] was sort of like this, leaning down, doing the thing, and I'm trying to get in on him to try to look over his shoulder," Hawn demonstrated. "Now I see the head and I'm going, 'Oh! There's the head!' Then it went back in and he said, 'Goldie, if you get any close, you're going to fall in!'"
Hudson welcomed Rani with Fujikawa in October. The couple named their baby girl after Fujikawa's late father, Ron Fujikawa, who died in 2012.
"Ron was the most special man who we all miss dearly. To name her after him is an honor," Hudson said at the time. "Everyone is doing well and happy as can be."
Hudson is also parent to two sons, 15-year-old Ryder with Chris Robinson and 7-year-old Bingham with Matt Bellamy. She is the daughter of Hawn and musician Bill Hudson.