Leigh was born Jennifer L. Morrow in Hollywood, Los Angeles, California, the daughter of Combat! actor Vic Morrow and Pollock screenwriter Barbara Turner. Both of Leigh's parents were Jewish, although Leigh was raised mostly without religion. Leigh changed her last name, taking the middle name "Jason" in honor of a family friend, Academy Award-winning actor Jason Robards.

At the age of 14, she attended Stagedoor Manor Performing Arts Training Center in Loch Sheldrake, New York and summer acting workshops given by Lee Strasberg. She received her Screen Actors Guild membership after appearing in her first credited role in an episode of the TV show Baretta when she was just shy of 15 years old.

An episode of The Waltons and several TV movies followed, including a portrayal of an anorexic teenager in The Best Little Girl in the World, for which Leigh dropped to 86 pounds (39 kg) under medical supervision. She made her screen debut as a blind, deaf, and mute rape victim in the 1981 slasher film Eyes of a Stranger, which she dropped out of high school to play.[citation needed] In 1982, she played a teenager who gets pregnant in Cameron Crowe's high-school comedy Fast Times at Ridgemont High, which was directed by Amy Heckerling and served as a launching pad for several then-unknown future stars, including Sean Penn, Judge Reinhold, Forest Whitaker, Eric Stoltz, Anthony Edwards, Phoebe Cates, and Nicolas Cage.

Leigh gravitated toward fragile, damaged, or neurotic characters. She was initially cast as victims – a virginal princess kidnapped and raped by mercenaries in Flesh & Blood (1985), an innocent waitress pursued by The Hitcher (1986), and a young woman on the verge of a nervous breakdown in the seedy nightclub inherited from her uncle in Heart of Midnight (1989).

It wasn't until 1990 that Leigh made a significant career breakthrough when she was voted the year's Best Supporting Actress by both the New York Film Critics Circle and the Boston Society of Film Critics for her portrayals of two very different prostitutes: first a tough streetwalker in Last Exit to Brooklyn, and then a sweet waif whose dreams of suburban bliss are shattered by sociopathic ex-con Alec Baldwin in Miami Blues. She then portrayed an undercover narcotics policewoman who becomes a junkie in the line of duty in Rush (1991), and followed that with one of her signature roles: Hedy, the psychotic “roommate from hell” in the thriller Single White Female (1992). She then played a fast-talking, hard-as-nails reporter who has her heart melted by Tim Robbins in the Coen Brothers’ surreal comic fantasy The Hudsucker Proxy (1994), and won many awards for her portrayal of the writer and poetess Dorothy Parker in Alan Rudolph's Mrs. Parker and the Vicious Circle (1994). For the latter, she received a Golden Globe nomination and Best Actress awards from the National Society of Film Critics, Chicago Film Critics Association and Fort Lauderdale Film Critics.

Next up was the role of Sadie Flood, an angry, drug-addicted barroom rock singer living in the shadow of her successful older sister (Mare Winningham) in Georgia (1995). For the role Leigh dropped to 90 pounds (41 kg) and sang all the songs live, including an 8½-minute version of Van Morrison's "Take Me Back". Peter Travers of Rolling Stone wrote that "[Leigh's] fierce, funny, exasperating and deeply affecting portrayal commands attention"; James Berardinelli claimed, "There are times when it's uncomfortable to watch this performance because it's so powerful", while Kenneth Turan of the Los Angeles Times said "Leigh’s exceptional performance tears you apart… we've never seen anything like it before." She won Best Actress awards from the New York Film Critics Circle and Montreal World Film Festival.

Other memorable Leigh roles of this era include a jaded phone-sex operator who diapers her newborn baby while plying her trade in Robert Altman's Academy Award-nominated film Short Cuts (1993), Kathy Bates's tormented, pill-popping daughter in the Stephen King adaptation Dolores Claiborne (1995), a streetwise kidnapper in Altman's jazz tribute Kansas City (1996), a mousy 19th-century spinster heiress courted by a gold-digger in Washington Square (1997), and a virtual reality game designer hunted by terrorists in David Cronenberg's surreal eXistenZ (1999). In 2001 she joined forces with Scottish actor Alan Cumming to write, direct, and produce a film shot in 19 days on digital video and starring real-life Hollywood friends such as Kevin Kline, Phoebe Cates, Gwyneth Paltrow, Jennifer Beals, John C. Reilly, and Parker Posey. The result was The Anniversary Party, a well-received ensemble comedy in the style of The Big Chill or Peter's Friends. Leigh and Cumming jointly received a citation for Excellence in Filmmaking from the National Board of Review and were nominated for two Independent Spirit Awards for Best First Feature and Best First Screenplay.

More recently Leigh has been cast in smaller character roles: gangster Tom Hanks's doomed wife in Sam Mendes's Road to Perdition (2002), Meg Ryan's brutally murdered sister in Jane Campion's In the Cut (2003), and Christian Bale's prostitute girlfriend in the thriller The Machinist (2004) (Mick LaSalle of the San Francisco Chronicle commented that "As the downtrodden, sexy, trusting and quietly funny prostitute, Leigh is, of course, in her element"). Her performance as a manipulative stage mother in Childstar won her a Genie Award in 2005.

Also a stage actress, Leigh took on the singing and dancing lead role of Sally Bowles in the popular musical Cabaret on Broadway from August 4, 1998, to February 28, 1999, and succeeded Mary-Louise Parker in the lead role in Proof from September 13, 2001, to June 30, 2002. Other theatrical appearances include The Glass Menagerie, Man of Destiny, The Shadow Box, Picnic, Sunshine, and Abigail's Party.

Leigh filmed a role for Stanley Kubrick's Eyes Wide Shut (1999), but when Kubrick wanted to do re-shoots, she was unavailable and her entire part was redone with actress Marie Richardson. In 1997, she was featured in Faith No More's music video for "Last Cup of Sorrow".