Sharp was classically trained in piano from an early age, taking up clarinet and guitar as a teen. He attended Cornell University from 1969 to 1971, studying anthropology, music, and electronics. He completed his B.A. degree at Bard College in 1973, where he studied composition with Benjamin Boretz and Elie Yarden; jazz composition, improvisation, and ethnomusicology with trombonist Roswell Rudd; and physics and electronics with Burton Brody. In 1977 he received an M.A. from the University at Buffalo, where he studied composition with Morton Feldman and Lejaren Hiller, and ethnomusicology with Charles Keil.[
From the late 1970s, Sharp established himself in New York's music scene. His compositions have been performed by the hr-Sinfonieorchester, the Ensemble Modern, Continuum, the Orchestra of the SEM Ensemble, Kronos Quartet, the FLUX Quartet, Zeitkratzer, the Soldier String Quartet, the JACK Quartet, and the Quintet of the Americas. His work has been featured at festivals worldwide, including the 2008 New Music Stockholm festival, the 2007 Hessischer Rundfunk Klangbiennale, and the Venice Biennale 2003 and 2006. He releases music under his own label (zOaR music) as well as punk label SST and downtown music labels such as Knitting Factory records and John Zorn's Tzadik label. Guitar Player magazine's 30th anniversary issue included Sharp among their list of "The Dirty Thirty - Pioneers and Trailblazers".
He has collaborated regularly with many people, including Christian Marclay, Eric Mingus, Zeena Parkins, Vernon Reid, Bobby Previte, Joey Baron, David Torn, Nels Cline, and Frances-Marie Uitti, as well as qawaali singer Nusrat Fateh Ali Khan, blues legend Hubert Sumlin, actor/writer Eric Bogosian, jazz greats Jack DeJohnette and Sonny Sharrock, pop singer Debbie Harry, and Bachir Attar, leader of the Master Musicians of Jajouka. He was curator of the sound-art exhibition Volume: Bed of Sound for P.S.1 Contemporary Art Center in 2001, which featured the works of 54 artists including Vito Acconci, Tod Dockstader, John Duncan, Walter Murch, Muhal Richard Abrams, Laurie Anderson, Chris & Cosey, Survival Research Laboratories, Ryuichi Sakamoto, Sonic Youth, and Butch Morris. He also curates the State of the Union CD compilations of one-minute tracks by experimental musicians, and produces records for a wide variety of artists. Sharp describes himself as a lifelong "science geek" having modified and created musical instruments since his teen years, and frequently borrowing terms from science and technology for his compositions.
Sharp has composed scores for feature films and documentaries; created music and sound-design for The Sundance Channel, MTV and Bravo networks; and has presented numerous sound installations in art galleries and museums.
He is the subject of Bert Shapiro's documentary Elliott Sharp: Doing the Don't, for Pheasant Eye Film. He was a 2009 Master Artist-in-Residence at the Atlantic Center for the Arts, and a 2010 New York Foundation for the Arts Fellow in music. In 2003 he was awarded a grant from the Foundation for Contemporary Arts Grants to Artists Award. In March 2011, Sharp's 60th birthday was celebrated with a weekend of all-star concert events entitled "E#@60", hosted by Brooklyn's ISSUE Project Room.
Sharp lives in lower Manhattan with designer/video artist Janene Higgins and their two children.