The couple became proud parents to a little boy last March (10), but kept the news secret for over two months.
They refused to divulge the tot's name in a bid to "protect" his privacy, but now Morgan has lifted the lid on life with the toddler, who is called Gus.
The actor says, "He just said 'no' for the first time, which is f**ked up. In one week it's just like, 'No, no, no!' But everything is a first. It's been spectacular."
Speaking to E! Online at the Los Angeles premiere of the new Twilight film on Monday (14Nov11), he adds, "(Life is) a lot of not sleeping, and then our one adult night we come here... this is our date. Can you believe it?"
Gus is Burton's first child and Morgan's second - the actor discovered he was a first-time father when his actress ex-girlfriend Sherrie Rose revealed in 2009 that her then four-year-old son was the result of her romance with the star.
The former One Tree Hill star, 28, began dating The Watchmen actor Morgan, 44, last year (09) but the private couple managed to keep the romance under wraps and tabloid reports in May (10) revealed the pair had quietly become parents.
Burton has now confirmed the happy news, telling the Associated Press, "(We are) really happy, and we're a very good team. And we got very lucky that our family is what it is."
But she insists on keeping the tot's name and birth date secret.
She says, "I want to protect them like a mama wolf should."
The child is Burton's first and Morgan's second - the actor discovered he was a first-time father after his actress ex Sherrie Rose revealed last year (09) that her then four-year-old son was the result of her romance with the star.
Burton was previously married to One Tree Hill assistant director Ian Prange but the couple split in 2008.