Golijov was born to Jewish immigrants from eastern Europe. He studied music with his mother, a piano teacher, and at the local conservatory in Argentina. He moved to Israel in 1983 and studied in Jerusalem, and in 1986 he moved to the United States, where he received a Ph.D. from the University of Pennsylvania (as a student of the American composer George Crumb (Crumb, George)). He studied further at Tanglewood, in Lenox, Mass., winning the Koussevitzky Composition Prize in 1990. A year later he joined the faculty at College of the Holy Cross in Worcester, Mass.; he was also on the faculty of the Boston Conservatory from 2001.

Golijov composed music in most principal genres. He commonly wrote for unconventional instruments (using the accordion, Cuban drums, cimbalom, or kamāncheh 【a Persian spike fiddle】, for example, with Western orchestral instruments), and he sometimes used sampled and amplified sounds. Called an art for the 21st century, his work drew on such influences as Jewish liturgical and folk music, particularly klezmer; popular Latin American dance forms; Middle Eastern songs; and Western classical music of all periods. His vocal works often employed languages, such as Yiddish and Arabic, that were unusual in Western classical music. Although his works, particularly for the voice, could be hauntingly lyrical, they were also often strongly rhythmic.

Golijov became identified with a number of prominent performers, including the St. Lawrence String Quartet, the Kronos Quartet, and the soprano Dawn Upshaw, and many of his works were developed in collaboration with such artists. One of his earliest successes, Yiddishbbuk (1992), was written for the St. Lawrence and the clarinetist Todd Palmer, and the Kronos Quartet performed and recorded a number of his works. In 2000 Golijov received acclaim for La Pasión según San Marcos, a Latin American setting of the Passion commissioned by the Bach Academy in Stuttgart, Ger., to commemorate the 250th anniversary of Johann Sebastian Bach (Bach, Johann Sebastian)'s death. Upshaw sang and recorded Ayre (2004), a cycle that included Arab, Hebrew, and Spanish songs, and she was featured on the Grammy Award-winning recording of Ainadamar, based on the murder of the Spanish playwright Federico García Lorca (García Lorca, Federico) and featuring a libretto by playwright David Henry Hwang. Golijov wrote original scores for films, including Sally Potter's The Man Who Cried (2000) and Francis Ford Coppola (Coppola, Francis Ford)'s Youth Without Youth (2007).

In 2006 New York City's Lincoln Center and London's Barbican Centre held major festivals of his music, and the Atlanta Symphony Orchestra, under Robert Spano, featured his music throughout the 2006–07 season. In September 2006 Golijov took up his two-season position as a composer in residence with the Chicago Symphony Orchestra. Golijov also served as composer in residence at the Spoleto Festival USA; the annual festivals held at Marlboro, Vt., and Ravinia, Ill.; the Mostly Mozart Festival at Lincoln Center; and elsewhere. His numerous honours include a MacArthur fellowship in 2003. In 2006 he was named Composer of the Year by the venerable performing-arts magazine Musical America.