The baby boy, who was born Dec. 25 in California via a surrogate, weighed 7 pounds, 15 ounces.
"We are overwhelmed with happiness and joy at this very special moment," John and Furnish tell Us in a joint statement. "Zachary is healthy and doing really well, and we are very proud and happy parents."
This is the first child for John, 62, and Furnish, 48. The couple married in 2005 after 12 years together.
A rep for the couple has stated that they intend to protect and respect the privacy of their surrogate, and will not be discussing any details relating to the surrogacy arrangements.