Orbach was born in the Bronx, a borough of New York City, to Emily Olexy (a Polish American Catholic) and Leon Orbach, a German Jew of Sephardic descent. He was raised as a Roman Catholic. While he was still a child, his family moved to Mount Vernon, New York; Wilkes-Barre and Scranton, Pennsylvania; Springfield, Massachusetts; and Waukegan, Illinois. He studied drama at Northwestern University, then went to New York, where he studied with Lee Strasberg at the Actors Studio.

Orbach was an accomplished Broadway and off-Broadway actor. His first major role was that of El Gallo in the original cast of the decades-running hit The Fantasticks. He also starred in Carnival!, the musical version of the movie Lili. He also starred in a revival of Guys and Dolls (Tony Award nomination for Best Featured Actor in a Musical), Promises, Promises (Tony Award for Best Actor in a Musical), the original productions of Chicago (Tony Award nomination for Best Actor in a Musical) and 42nd Street, and a revival of The Cradle Will Rock.

In the 1980s, he shifted to film work, including prominent roles as Jennifer Grey's father in Dirty Dancing, a cold-blooded killer in the Woody Allen drama Crimes and Misdemeanors, and the voice of the candelabra Lumiere in Disney's animated musical Beauty and the Beast. He starred in the short-lived 1987 crime drama The Law and Harry McGraw (playing a role that he originated and later reprised as a regular guest star on Murder, She Wrote for several years), which foresaw his best-known role of all, that of Detective Lennie Briscoe in the series Law & Order (1992-2004). Orbach also voice acted the character for the video game spin-offs of the series. Orbach was signed to continue in the role on Law & Order: Trial by Jury. He appeared in only the first two episodes of the series, which aired in March 2005, after his death. The fifth episode of the series, "Baby Boom", was dedicated to his memory.

In early December 2004, it was announced that Orbach had been receiving treatment for prostate cancer since Spring 2004; he died at the Memorial Sloan-Kettering Cancer Center in New York on December 28. His agent, Robert Malcolm, announced at the time of his death that Orbach's prostate-cancer had been diagnosed more than ten years before.

Orbach was married in 1958 to Marta Curro, with whom he had two sons, Anthony Nicholas and Christopher Benjamin; they divorced in 1975. In 1979, he married Broadway dancer Elaine Cancilla, whom he met while starring in Chicago. In addition to his sons and both wives, Orbach was survived by his mother.

He was named a "Living Landmark", along with fellow castmate Sam Waterston, by the New York Landmarks Conservancy in 2002. He quipped that the honor meant "that they can't tear me down".

Orbach lived in a high-rise off Eighth Avenue in Clinton and was a fixture in that Manhattan neighborhood's restaurants and shops. His glossy publicity photo hangs in Ms. Buffy's French Cleaners, and he was a regular at some of the unpretentious Italian restaurants nearby.

On February 5, 2005, he was posthumously awarded a Screen Actors Guild Award for Outstanding Performance by a Male Actor in a Drama Series.