Jenkins was born in 1885 in Jenkinsburg, Georgia, on the edge of Atlanta. He was left partially blind by a mis-prescribed medication while still an infant. Early on, he exhibited remarkable musical talents and was able to play almost any instrument he picked up, learning completely by ear. Jenkins, who also showed a proficiency for writing songs at a young age, saw his musical abilities as "a God gift."

Because he had some vision, Jenkins could not attend the state's school for the blind and had to pursue an education on his own. Besides his musical talents, he was skilled in other ways. After becoming a Methodist at the age of nine, he began "preaching" to playmates from porches and tree stumps. Not surprisingly, he became a licensed preacher around the age of 21 and moved into the city, supplementing whatever he could earn from preaching and street performing by running a newspaper stand. After his first wife's death, Jenkins married Francis Jane Walden Eskew in 1919. A young widow, his new wife had three musically talented children, Irene, Mary Lee and a son, T.P. Thus was born the Jenkins Family, one of the most popular family acts of its day.

Little is known about the Jenkins' musical development over their first three years together, but in 1922, they performed their first program on Atlanta radio station WSB with Andrew Jenkins billed as "the blind newsboy evangelist." The station, which had begun broadcasting just five months before and had a signal that reached coast-to-coast, became known as The Voice of the South. Performing folk, country and light classical material, the Jenkins Family was an immediate success and remained with the station for nearly a decade. Their popularity, which reached to Canada and Mexico, also attracted the attention of a major record label, Okeh Records, for whom they made their debut recordings in 1924 (three years before the Carter Family began recording).

Jenkins was adept at more than just ballads, turning out one song after another, as his stepdaughter Irene described it, like a mill grinding wheat. Many of his topical compositions were recorded by other artists over the ensuing years, including "Billy the Kid", drawn from a popular 1926 biography of the Cowboy killer; "The Wreck of the Royal Palm", about a two-train collision in Georgia in 1926; "The Tragedy on Daytona Beach", based on the 1929 death of race car driver Lee Bible in Florida; "The Fate of Frank Dupre", the story of an Atlanta robbery/murder; and "Ben Dewberry's Final Run", which recounted the death of an Atlantan who was the Engineer in a 1908 train wreck. In all, he wrote more than 300 non-sacred songs, many of them "news ballads".

Rev. Jenkins and the Jenkins Family continued to record with Okeh into the mid-1930s under a variety of names, including the Jenkins Sacred Singers, Irene Spain Family, Gooby Jenkins, Andrew Jenkins & Carson Robison and Jenkins & (Byron) Whitworth. Jenkins cut his last record as an accompanied soloist billed as Blind Andy, a name he often recorded under, on April 23, 1930, while the Jenkins Family's last recording session for Okeh was July 30, 1934.

The Jenkins Family's popularity opened Rev. Jenkins's ministry to a wide audience, as they performed at concerts and organized revival meetings throughout Georgia. Over time, Jenkins also served as pastor of several churches. In 1939, he lost his eyesight completely, yet continued to preach until the time of his death and also broadcast as an evangelist on Mexico's powerful, uncontrolled radio stations.

Jenkins was killed in an automobile accident in 1957.

Although the genre's popularity lasted just one brief year, country artists returned to the form again and again – in the late 1920s, pioneers such as Jimmie Rodgers and the Carter Family and in the 1930s, early folk artists such as Woody Guthrie and Lead Belly. The genre was revisited during the folk revival of the 1950s and 1960s, in particular, by Bob Dylan, who wrote one of his first songs based on the same "formula" used by Jenkins, and Phil Ochs, many of whose songs came directly from newspaper headlines.