The 33-year-old gave birth to a son, Cyrus Michael Christopher Dancy, on Monday.
It's the first child for the 'Homeland' star and her British actor hubby, Dancy who married in 2009.
Hugh and a pregnant Claire at this year's Emmys
It tops off a huge year for the actress who was nominated for a Screen Actors Guild Award and a Golden Globe for her role in 'Homeland'.
She's up for the Best Performance by an Actress in a Television Series gong at the Globes ceremony on 13 January.