Shaw was born in Greenville, North Carolina and began playing the violin when she was two years old. Her mother was her first teacher. She began writing music when she was ten years old, mostly in imitation of the chamber music of Mozart and Brahms. However, her main focus was then on violin performance. Shaw received her Bachelor of Music (violin performance) from Rice University in 2004, and her master's degree (violin) from Yale University in 2007. She entered the PhD program in composition in Princeton University in 2010.
At 30, Shaw became the youngest recipient of the Pulitzer Prize for Music for her composition "Partita for 8 Voices". The jury citation praised the composition as "a highly polished and inventive a cappella work uniquely embracing speech, whispers, sighs, murmurs, wordless melodies and novel vocal effects." The work comprises four movements inspired by the baroque dance forms: Allemande, Sarabande, Courante and Passacaglia. A recording of the work has been released by New Amsterdam Records on 30 October 2012, performed by the ensemble Roomful of Teeth. According to Steven Mackey, chair of the Department of Music at Princeton, this is the first Pulitzer Prize awarded to a member of the department. (Milton Babbitt was awarded a Pulitzer citation in 1982 for his life's work as a composer.)
Besides composition, Shaw is known as a musician appearing in many guises. She performs primarily as violinist with the American Contemporary Music Ensemble (ACME) and as vocalist with Roomful of Teeth. She also works with the Trinity Wall Street Choir, Alarm Will Sound, Wordless Music Orchestra, Ensemble Signal, AXIOM, The Yehudim, Victoire, Opera Cabal, the Mark Morris Dance Group Ensemble, Hotel Elefant, the Oracle Hysterical, Red Light New Music, Robert Mealy's Yale Baroque Ensemble.
Her works have been performed by Roomful of Teeth, So Percussion, ACME, and the Brentano String Quartet. Shaw has been a Yale Baroque Ensemble fellow and a Rice University Goliard fellow. She received the Thomas J. Watson Fellowship in 2004/5.