Kavafian was born in Istanbul to parents of Armenian descent. In 1956 she moved with her family to Detroit, by which time she had begun playing piano. She began on violin at age nine, studying under Ara Zerounian and Mischa Mischakoff. She studied at Juilliard from 1966 under Ivan Galamian and Sally Thomas. She debuted at Carnegie Hall in 1969 and in Paris in 1973, the same year she won the Young Concert Artists International Auditions. Her sister, Ida Kavafian, also studied at Juilliard, and the two have played together frequently in concert over the course of their careers. Ani toured through the 1970s, and taught at the Manhattan School of Music and the Mannes College of Music in the 1980s. She became a member of the Chamber Music Society of Lincoln Center in 1979, and married artist Bernard Mindich. She has also taught and performed at the Sarasota Music Festival.
Kavafian's repertory is grounded in the concertos of Mozart, Brahms, and Mendelssohn, but she is also well known for her performances of twentieth century works of Samuel Barber, Béla Bartók, Dmitri Shostakovich, Erich Korngold, Aram Khachaturian, Arno Babajanian, Henri Lazarof, and Tod Machover. She plays a 1736 Stradivarius.