Veracini was born in Florence, Italy, the eldest son of Francesco di Niccolò Veracini, a noted violinist who ran a music school, and from whom Antonio first learned to play the violin. When his father's health began to fail around 1708, Antonio took over the running of the school, where he taught the violin to (amongst others) his nephew Francesco Maria Veracini (1690–1768), later a celebrated violinist and composer in his own right. Unlike his nephew, who travelled widely, Antonio rarely left Florence. He did visit Rome on two occasions, where he is believed to have met Arcangelo Corelli, and in 1720 he briefly visited Vienna (Hill 2001).
Antonio Veracini died aged 75, which for the time period was a ripe age, but not particularly old.