A source said: "Everyone is healthy and they couldn't be happier."
Exton weighed in at 7lb 5oz and measured 20 inches long.
Robert, 46, had previously revealed on live television he and Susan were expecting a son, much to his producer wife's shock.
She later said: "I could never really stay mad at him, but I can tell you it genuinely wasn't something we planned to reveal."
However, the 'Sherlock Holmes' star did manage to keep quiet about their name choices for the baby.
He joked recently: "If I told you the name, then she would lop off my head with a machete."
The actor recently admitted impending fatherhood was making him consider slowing down - though he admitted it was unlikely to happen.
He said: "Life is good. With a new baby on the way, maybe I ought to think about slowing down a little. But I can't see that happening in the short term."
