Jost announced the news on his Instagram Wednesday.
"Ok ok we had a baby. His name is Cosmo. We love him very much," Jost wrote.
No details as to when the baby was born were shared. He went on to say "privacy would be greatly appreciated."
This is the second child for Johansson, who has daughter Rose Dorothy, 7, with her ex-husband Romain Dauriac.
The actress and "SNL" star got engaged in 2019 and wed in 2020.
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