Gilbert was born in New York in 1967. Both his parents were violinists in the New York Philharmonic. He studied at Harvard University, the Curtis Institute of Music, and the Juilliard School, and won the Georg Solti Prize in 1994. From 1995 to 1997, he served as assistant conductor with The Cleveland Orchestra.

From 2009 to 2017, Gilbert was music director of the New York Philharmonic. He introduced the CONTACT! new-music series and the NY PHIL BIENNIAL, and led staged productions of works by Ligeti, Janáček, and Stravinsky.

Gilbert regularly conducts major orchestras including the Berliner Philharmoniker, The Cleveland Orchestra, Boston Symphony Orchestra, Symphonieorchester des Bayerischen Rundfunks, and Gewandhausorchester Leipzig. He has led opera productions in New York, Milan and Dresden, as well as Santa Fe Opera, where he was music director from 2003 to 2006.

His recording of John Adams's Doctor Atomic for the Metropolitan Opera won a Grammy Award. He was appointed Royal Court Kapellmeister by the King of Sweden in 2022 and named an Officier de l'Ordre des Arts et des Lettres by the French government.

Gilbert made his debut with the Royal Concertgebouw Orchestra in 2003 and most recently appeared in 2021, conducting works by Rachmaninoff, Nielsen, and Richard Strauss.