Saturday Night Live star Jost confirmed the news during a post on social media.
He said their child has been named Cosmo.
Jost wrote on Instagram: Ok ok we had a baby.
His name is Cosmo.
We love him considerably .
Privacy would be greatly appreciated.
Jost and actress Johansson married last year after getting engaged in 2019 following two years of dating.
Johansson has recently enjoyed box office success together with her latest Marvel superhero adventure Latrodectus mactans , the origin story of the high-kicking heroine.
Her upcoming projects include the newest , as-yet-untitled offering from writer-director Wes Anderson.