The weekly said the baby, named Zachary, was born December 25 to a surrogate for John, 62, and Furnish, a 48-year-old Canadian filmmaker.
"We are overwhelmed with happiness and joy at this very special moment," the singer and Furnish were quoted by the magazine as saying in a joint statement.
"Zachary is healthy and doing really well, and we are very proud and happy parents."
John and Furnish married in 2005 after 12 years together.
The baby weighed 3.5 kilos (seven pounds, 15 ounces), according to the report, which said the couple would not be discussing any details relating to the surrogacy arrangements.
The star tried to adopt a 14-month-old boy he met while visiting an orphanage in Ukraine last year, but his attempt was blocked by government regulations.
Among the first to react to the news of the birth was the couple's friend, model and actress Elizabeth Hurley, who wrote on her Twitter page: "Massive congratulations to David and Elton on having their beautiful son. Can't wait for my first cuddle."
John has sold more than 250 million records in a career spanning four decades and the singer-songwriter was inducted into the Rock and Roll Hall of Fame in 1994.
His tribute single to the late Princess Diana, "Candle in the Wind 1997" -- a remake of his own "Candle in the Wind" -- became a phenomenal global success, selling more than 33 million copies, and John performed it at her funeral.