It's their first child together, although the 46-year-old actor already has a 18-year-old son from his previous marriage to Deborah Falconer.
The Avengers star's son has been named Exton Elias and was born on Tuesday morning, weighing 7lbs 5oz.
Robert previously announced the sex of the baby live on TV, a declaration which upset his wife.
However, Susan later said: “I couldn’t stay mad at him, but I can tell you it genuinely wasn’t something we planned to reveal”.
He did however manage to keep the name under wraps.
“If I told you the name, then she would lop off my head with a machete”, joked the Sherlock Holmes actor recently.
According to a source both mother and son are healthy and the family couldn’t be happier.