Quinn was born in Rockford, Illinois to devoutly Catholic Irish parents. He was raised in Chicago and Rockford, as well as in Dublin, in Birr in County Offaly, and other places in Ireland. His father was a professor in literature and his mother a homemaker.

He has three brothers and a sister. His older brother, Declan Quinn, is a noted cinematographer. Though a roofer by trade, Quinn got his start in the Chicago theatre at age 19. He trained at the Piven Theatre Workshop.

His first significant film role was in Reckless, followed by a breakthrough role in Desperately Seeking Susan as the character "Dez" (the love interest of the character played by Rosanna Arquette).

Quinn next starred in the controversial TV movie An Early Frost, about a young lawyer dying of AIDS (it was broadcast on NBC on November 11, 1985) and co-starred such luminaries as Gena Rowlands, Ben Gazzara and Sylvia Sidney. He received his first Emmy Award nomination for the role which allowed him to gain recognition in Hollywood.

In 1988 Quinn lost the role of Jesus Christ when Paramount Pictures dropped the controversial Martin Scorsese movie The Last Temptation of Christ. When Universal Pictures picked up the film, the role went to Willem Dafoe.

During the 1990s Quinn's career grew immensely. He found work alongside such Hollywood stars as Brad Pitt in Legends of the Fall, Johnny Depp in Benny & Joon and Robert Duvall in The Handmaid's Tale. He continually starred in films that allowed him to return to his Irish roots, such as Michael Collins and Evelyn

In 2000, Quinn portrayed Paul McCartney in the VH1 television drama, Two of Us. He later met McCartney at a hotel, and they became good friends. Quinn was able to gain his first regular television role as the lead in the short lived, and controversial NBC drama The Book of Daniel, in 2006 before being cancelled after the first three weeks of its run.

In 2007, Quinn received his second Emmy nomination for the television movie Bury My Heart at Wounded Knee.