Abraham was born Murray Abraham on October 24, 1939 in Pittsburgh, Pennsylvania, the son of Fahrid “Fred” Abraham, an auto mechanic, and his wife Josephine (née Stello) (April 15, 1915 – March 10, 2012), a housewife. His father emigrated with his family from Ottoman Syria at age five due to the famine of Mount Lebanon; his paternal grandfather was a priest in the Antiochian Orthodox Church His mother, one of 14 children, was Italian American, and the daughter of an Italian immigrant who worked in the coal mines of Western Pennsylvania. He had two younger brothers, Robert and Jack, who were killed in separate car accidents.

Abraham was raised in El Paso, Texas, near the Mexican border. Murray and his two younger brothers were altar boys in the St. George Antiochian Orthodox Church in El Paso. He attended Vilas Grammar School, and graduated from El Paso High School in 1958. He was a gang member during his teenage years. In El Paso, Abraham worked in the Farah Clothing factory owned by a Lebanese family before launching a career in acting. He attended Texas Western College (later named University of Texas at El Paso), where he was given the best actor award by Alpha Psi Omega for his portrayal of the Indian Nocona in Comanche Eagle during the 1959–60 season. He attended the University of Texas at Austin, then studied acting under Uta Hagen at HB Studio in New York City. He began his acting career on the stage, debuting in a Los Angeles production of Ray Bradbury’s The Wonderful Ice Cream Suit.

Despite these small roles, Abraham continued to do commercials and voice-over work for income. But in 1978, he decided to give them up. Frustrated with the lack of substantial roles, Abraham said, “No one was taking my acting seriously. I figured if I didn’t do it, then I’d have no right to the dreams I’ve always had.” His wife, Kate Hannan, went to work as an assistant and Abraham became a “house husband”. He described, “I cooked and cleaned and took care of the kids. It was very rough on my macho idea of life. But it was the best thing that ever happened to me.”

Abraham gained greater prominence when he appeared as drug dealer Omar Suárez in the gangster film Scarface (1983). Then, in 1984, he played envious composer Antonio Salieri in the Academy Award for Best Picture-winning Amadeus (1984), directed by Miloš Forman. Abraham won the Academy Award for Best Actor for his role, an award for which his co-star in the film Tom Hulce, playing Mozart, had also been nominated. He won a Golden Globe Award, among other awards, and his role in the film, remains as his most iconic. He later continued his association with classical music by narrating the plot summaries of the operas of Wagner’s Ring Cycle in the 1990 PBS broadcast from the Metropolitan Opera, to the largest viewing audience of the Ring Cycle in history, conducted by James Levine.

Abraham also joined The Mirror Theater Ltd’s Mirror Repertory Company in 1984. He joined MRC the week after winning his Oscar for Best Actor for his work in Amadeus because he wanted to work with MRC Artist-in-Residence Geraldine Page (to whom he would eventually present her own Academy Award the following year), and would star opposite her in MRC’s The Madwoman of Chaillot.

In 1994, Abraham portrayed Roy Cohn in the first Broadway production of Tony Kushner’s Angels in America at the Walter Kerr Theater, replacing Ron Leibman in the role.

In July 2004, during a ceremony in Rome, he was awarded the “Premio per gli Italiani nel Mondo”. This is a prize distributed by the Marzio Tremaglia foundation and the Italian government to Italian emigrants and their descendants who have distinguished themselves abroad.

Abraham has focused on stage work throughout his career, giving notable performances as Pozzo in Mike Nichols’s production of Waiting for Godot, Malvolio in Twelfth Night for the New York Shakespeare Festival, and Shylock in The Merchant of Venice for the Off-Broadway Theatre for a New Audience (TFANA) in March 2007, which was performed at the Duke Theatre in New York and also at the Swan Theatre, part of the Royal Shakespeare Company. He reprised this role in February 2011, when he replaced Al Pacino in the Public Theater’s production. In the 1997/98 Broadway season, he starred in the new chamber musical Triumph of Love opposite Betty Buckley, based on Marivaux’s classic comedy. The production did not find a large audience, running 85 performances after its pre-opening preview period. He has also taught theater at Brooklyn College. In 2016, he played the title role in Classic Stage Company’s production of Nathan the Wise.

A 2009 guest appearance on Saving Grace began a new phase of Abraham’s career, wherein he has become gradually more prolific onscreen. Further guest appearances include roles on Law & Order: Criminal Intent, Louie and Curb Your Enthusiasm as well as a recurring role on The Good Wife between 2011 and 2014. Additionally, Abraham was the primary narrator for the PBS series Nature between 2007 and 2010, narrating 32 episodes (plus one more in 2013). Abraham’s most notable television role came about through Showtime’s drama series Homeland, in which he portrayed black ops specialist Dar Adal. This role resulted in his first Emmy Award nomination in 2015, followed by a second in 2018.

In 2009, he was recognized by the Alumni Association of the City College of New York with John H. Finley Award in recognition of exemplary dedicated service to the City of New York.

In January 2010, Abraham was the on-the-scene hero of a real-life crime scene at the Classic Stage Company in New York City, when he scuffled with a thief in the dressing room area during a public rehearsal.

In 2010, Abraham was the recipient of The Gielgud Award (Theatre) for that year.

In 2015, Abraham was an inductee to the American Theater Hall of Fame.

In the 2010s, he has featured prominently in two widely acclaimed films: first as folk music impresario Bud Grossman in the Coen brothers’ drama Inside Llewyn Davis (2013), then as the mysterious Mr. Moustafa in Wes Anderson’s The Grand Budapest Hotel (2014). More recently, he has voiced roles in Isle of Dogs (2018) and How to Train Your Dragon: The Hidden World (2019) and played Tony in the 2019 live-action Lady and the Tramp.