Melissa Aldana was born in Santiago, Chile. She began playing the saxophone when she was six, under the influence and tuition of her father Marcos Aldana, also a professional saxophonist. Aldana began with alto, influenced by artists such as Charlie Parker, Cannonball Adderley, and Michael Brecker. However, upon first hearing the music of Sonny Rollins, she switched to tenor; the first tenor saxophone she used was a Selmer Mark VI that had belonged to her grandfather.
She started performing in Santiago jazz clubs in her early teens. In 2005, after meeting him while he was on tour in Chile, she was invited by pianist Danilo Pérez to play at the Panama Jazz Festival, as well as a number of auditions at music schools in the USA. As a result of these introductions, she went on to attend Berklee College of Music in Boston, where her tutors included Joe Lovano, George Garzone, Frank Tiberi, Greg Osby, Hal Crook, Bill Pierce, and Ralph Peterson
Aldana graduated from Berklee in 2009, relocating to New York City to study under George Coleman. She recorded her first album, Free Fall, released on Greg Osby’s Inner Circle Music imprint in 2010. Her live shows in this period included performances at the Blue Note Jazz Club and the Monterey Jazz Festival, and her second album, Second Cycle, was released in 2012. In 2013, aged 24, she was the first female musician and the first South American musician to win the Thelonious Monk International Jazz Saxophone Competition, in which her father had been a semi-finalist in 1991. The prize was a $25,000 scholarship, and a recording contract with Concord Jazz. Reporting her win, the Washington Post described Aldana as representing «a new sense of possibility and direction in jazz».