"We are overwhelmed with happiness and joy at this very special moment," John, 63, and Furnish, 48, told Us magazine in a joint statement.
"Zachary is healthy and doing really well, and we are very proud and happy parents."
The 7-pound, 15-ounce boy is a first child for John and Furnish, who have been together since 1993 and certified their relationship in a 2005 civil partnership ceremony.
The surrogate was not identified, and a representative for the couple told the magazine that for the surrogate's privacy they would not discuss details.
John and Furnish, who produces the Sundance Channel music interview / performance series "Spectacle: Elvis Costello with . . . " are the latest in high-profile gay fathers who have had children via surrogacy.
And while Zachary Jackson Levon was not "born a pauper to a pawn on a Christmas day," like the Levon in John's 1971 hit of that name, his parents will surely "send him to the finest school in town."