The couple gave birth to the baby girl on Friday September 12, according to US media.
The couple had also kept the news of Mendes's pregnancy to themselves for a long time, with the first reports emerging in July when she was already seven months pregnant.
Mendes, 40, has not been seen on the red carpet since February and had brushed off pregnancy rumours earlier this year.
At the time she said: "It's so ridiculous.
"It all started because I didn't want to go through the scanners at the airport. You know those X-ray scanners which are really creepy? They basically see you naked, right?
"And, not only that, but there's a radiation aspect to it so I always opt out."
Mendes, who made her name in the 2001 blockbuster Training Day, has been dating The Notebook star Gosling, 33, for three years.
Both played parents in the film The Place Beyond The Pines last year.
you're the one who accepts roles that put you in this rut or in this stereotype."