Morrison was born in New York City, the only daughter of Sephardic Jewish (specifically, Spanish Jewish) parents, originally surnamed Mitrani. She was raised in the Bronx, New York, and her first language was Spanish. Her father, Maurice Morris, was a clothing manufacturer. Morrison moved to California with her parents in 1946.

From 1965 to 1967, she appeared four times as an American Indian, Linda Little Trees, in the NBC western television series Laredo (in the episodes "Yahoo", "Jinx", "No Bugles, One Drum", and "Split the Difference.") In 1967, she capitalized on her own experiences learning English to play the Puerto Rican nun, Sister Sixto, who had trouble with English in Sally Field's show The Flying Nun. She would keep this role until the show was cancelled in 1970.

Morrison underwent a lumpectomy for breast cancer in 1988, followed by a full mastectomy when the cancer recurred in 1998. She had lung cancer and underwent surgery to remove the upper third of her right lung. Morrison's lung cancer went into remission, but her breast cancer returned after the treatment in 1988 and 1998.

Morrison's most widely recognizable recent role has been as the Salvadoran maid Rosario Inés Consuelo Yolanda Salazar on the NBC sitcom Will & Grace. She appeared in 68 episodes from 1999 to 2006 (as well as a brief cameo appearance on the September 2016 Will & Grace webisode in support of Hillary Clinton's presidential campaign). The role was originally created for a brief one-episode appearance, but Rosario was so popular with viewers that she became a recurring character. In addition to sparring with her libertine and very spoiled employer, Karen Walker (Megan Mullally), Rosario married Karen's gay friend Jack McFarland (Sean Hayes) to prevent her impending deportation.

Morrison endorsed U.S. Congressman Dennis Kucinich for President in 2008, actively fundraising, contributing to his campaign, and seeking votes for him.

Morrison lives with her husband of many years, Walter Domínguez, a Mexican-American, in a three-story apartment in Los Angeles – the same apartment in which her father had settled the family five decades earlier. Morrison and Dominguez adopted three sons and three daughters (adopted through a traditional Native American ceremony).

Among approximately 25 film appearances and 200 television appearances, Morrison has portrayed a maid or housekeeper on 32 separate occasions. Morrison had just informed her agent not to offer her any more "maid parts" when the call came for Will & Grace. In 2017, NBC revived Will & Grace for a new series of episodes. On August 3, 2017, co-creator Max Mutchnick announced to reporters that Morrison had been asked to reprise her role as Salazar on the show, but that she had ultimately decided to retire completely from acting. Although her last official acting role was a voiceover in the 2012 animated film Foodfight!, her final appearance as an Actress was the 2016 election-themed webisode of Will & Grace.