Born and raised in San Francisco, California, she was a member of the second class of women to graduate from Yale. Afterwards, she went on to earn a master’s degree from the School of Oriental and African Studies, London University. While living in London she began to work for NBC News in their London bureau. Among her early reports was an interview with Libyan leader Muammar Qaddafi.

In 1978 Cockburn moved to CBS. As a New York-based producer for the network in the 1980s she covered, among other topics, the U.S.-directed Contra War against Nicaragua. Her 1984 report, “The Dirty War,” for which she traveled through regions of Nicaragua that were officially off-limits as being too dangerous for journalists to visit, revealed the Contras' horrifying record of routine atrocities against the civilian population. In subsequent reports she laid out the degree to which Contras were heavily involved in the narcotics business as well as the first full account of the role of White House aide Colonel Oliver North in directing the whole Contra war.

Following the overthrow of the Duvalier regime in Haiti in 1986, Cockburn’s report “Haiti’s Nightmare” (1987) on the brutality of a Haitian military unit being armed and trained by the U.S. led to an outcry in Congress and the suspension of all U.S. military aid to Haiti.

Other stories covered by Cockburn in this period included Pentagon military procurement scandals and the political history of then-Senator Jesse Helms in North Carolina. Shortly afterwards, Helms launched a campaign for his right-wing followers to buy the network.

In 1987 Cockburn began producing and reporting documentaries for PBS Frontline in collaboration with her husband Andrew Cockburn. In Guns, Drugs, and the CIA, (1987) she interviewed, on camera, Tony Po, aka Anthony Posephne, a legendary covert operations officer who had supervised the CIA’s secret war in Northern Laos during the 1960s and early 1970s. In the interview, Po stated that the CIA had supplied air transport for the heroin shipments of their local ally, General Vang Pao, the only such on-the-record confirmation by a former CIA officer concerning agency involvement in the narcotics trade.

In 1990 Cockburn produced and directed “From the Killing Fields” for the ABC News documentary show Peter Jennings Reports. The film revealed that the U.S. had long been covertly supporting the Khmer Rouge, the genocidal movement responsible for the deaths of millions in Cambodia in the 1970s who had been displaced by a Vietnamese-back regime in 1979. U.S. assistance to the murderous group, Cockburn revealed, had been ongoing throughout the 1980s. Following her report, an embarrassed Bush Administration terminated the covert program, a move that led to the eventual UN-supervised peace settlement in Cambodia.

During the Gulf War in 1991, Cockburn reported from Israel on the Iraqi Scud attacks against Tel Aviv. Her film, shot from a high rise building close to the impact zone, provided irrefutable evidence that contrary to official reports, the U.S.-supplied Patriot missiles were not only entirely failing to intercept the Scuds but were instead impacting on the city itself. Her 1991 PBS Frontline documentary The War We Left Behind, produced with her husband Andrew Cockburn, exposed the disastrous impact of economic sanctions on ordinary Iraqis and helped persuade the Vatican take a stand against the sanctions policy.

In 1997, Cockburn conceived and co-produced The Peacemaker, starring George Clooney and Nicole Kidman, a thriller positing a terrorist attack on New York City with a stolen nuclear weapon. In a 60 Minutes report produced the same year, the former Russian National Security Adviser, General Alexander Lebed, admitted that several “nuclear suitcases” in the Russian inventory had gone missing.

In 1998, Cockburn served as Ferris Professor of Journalism at Princeton University.

After teaching for a semester, Cockburn returned to full time journalism, producing a number of pieces for 60 Minutes. In 2000, she produced "America's Worst Nightmare," a 60 Minutes report on political instability in nuclear-armed Pakistan and the growing power in the country of fundamentalist groups linked to the Taliban, a piece that was recognized as "strikingly prophetic" in receiving the Alfred I. duPont-Columbia University Award in 2001. In 2002 she did a report on the true effects of U. S. anti-narcotics aerial spraying on the civilian population of Colombia.

In 2009, Leslie Cockburn directed and co-produced (with husband Andrew Cockburn) her first feature documentary for theatrical release. American Casino relates the story of the origins, progress and consequences of the subprime mortgage disaster that led to the greatest financial crisis since the Great Depression. Beginning filming in January 2008, Cockburn laid bare both the financial machinations and miscalculations on Wall Street that produced the disaster as well as its effects on an array of Baltimore homeowners struggling to stay afloat. The film premiered at New York's Tribeca Film Festival in April 2009.

Variety described it as a " searing expose of the subprime mortgage crisis (matching) Wall Street's numbers and graphics to the flesh-and-blood individuals whose lives have been devastated by the deliberate machinations of bankers and traders."

Slant magazine called it a "revelatory howl against the still-gestating, $8 trillion-and-counting financial-services industry bailout."

Nobel Prize prizewinning economist Joseph Stiglitz said“American Casino is a powerful and shocking look at the subprime lending scandal. If you want to understand how the US financial system failed and how mortgage companies ripped off the poor, see this film.”

In its review of the film, The New York Times noted that the film "does not leave you assured the worst is over.... The lesson of this story: If enough money is involved, greed trumps morality."

In the course of her career, Cockburn has won numerous awards including The Hillman Prize in 1983, the George Polk in 1990, and 2009 Columbia Dupont in 2002, Overseas Press Club, and 1991 Robert F. Kennedy Journalism Award.

She lives in Washington, D.C. with her husband, Andrew Cockburn, a journalist and film producer with whom she has co-authored several books. Together they have three children, Chloe Frances Cockburn, The O.C. and House, M.D. actress Olivia Wilde, and Charles Philip Cockburn.

Cockburn has two brothers in law, Alexander Cockburn and Patrick Cockburn, who are also journalists, and the mystery writer Sarah Caudwell was her half-sister in law. Her son-in-law is filmmaker Tao Ruspoli, and journalists Laura Flanders and Stephanie Flanders are her half-nieces by marriage, daughters of her half-brother in law Michael Flanders. Her father in law was Claud Cockburn.