It is the couple's first child who was born over the weekend, multiple sources confirm to People magazine.
Dunst and Plemons got together back in 2016, after meeting on the set of season two of their FX series. They have since remained very private about their relationship and news of their engagement only became public when Dunst wore an engagement ring to the Golden Globes in 2017.
Neither Kirsten or her Fargo co-star Jesse's representatives have commented on the reports as yet.
Speaking about her plans for the future, she said at the time: "I'm at a stage in my life where I'm like, I've been working since I was three".
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"It's time to have babies and chill", the actress said.
The 35-year-old first announced she was pregnant via a stunning photoshoot for Rodarte where she looked like an absolute Earth goddess. I love her so much.
We're so happy for Dunst and Plemons!
'That love is just like... you can't experience that unless you have a kid'. "We're having a really good time".