Morgan was born in Mancos in Montezuma County in far southwestern Colorado. Her family moved to California by the time she was in high school. Morgan had six siblings; five brothers and one sister. In the late 1940s, at Verdugo Hills High School in the Tujunga neighborhood of Los Angeles, she served as class treasurer (and got the nickname "Jaye P." after the banker J. Pierpont Morgan) and sang at school assemblies, accompanied by her brother on guitar.
In 1950, a year after her graduation from high school, Morgan made a recording of "Life Is Just a Bowl of Cherries" issued by Derby Records, which made it to #26 in the U.S. Billboard record chart . Soon after, she received an RCA Victor recording contract, and she had five hits in one year, including "That's All I Want from You," her biggest hit, which reached #3 on the chart. Other notable hits included "The Longest Walk" and "Pepper Hot Baby". In 1954, she married Michael Baiano. She joined MGM Records in 1959 after spending the previous six years with RCA Victor.
From 1954 to 1955, Morgan was a vocalist on the ABC television series show Stop the Music. In November 1955, the British music magazine, NME, reported that Morgan was the top female vocalist in the U.S. Cash Box poll.
She spent considerable time in the 1960s making nightclub appearances. In 1966, she guest starred on CBS's My Three Sons as fading singer Claudia Farrell in the episode entitled "A Falling Star".
In 1961, Morgan was cast as Sally Dwight in the episode "Money and the Minister" of the CBS anthology series, General Electric Theater, hosted by Ronald Reagan. In 1962, she played Patty Maxwell in "Patti's Tune" of the CBS military sitcom/drama Hennesey, starring Jackie Cooper. That same year, she was cast as Kitty Flanders in "That's Showbiz" on NBC's The Joey Bishop Show. In 1964, Morgan was cast as Ruth Evans in the episode "Sunday Father" of the NBC medical drama, The Eleventh Hour.
In 1973, Morgan played herself in the episode "The Songwriter" of the sitcom, The Odd Couple. She appeared as Magda Valentine in the film The All-American Boy, and continued to play small roles in films such as Loose Shoes (1980), Night Patrol (1984) and Home Alone 2: Lost in New York (1992).
From 1976 to 1978, Morgan was a regular panelist on the popular game/variety show The Gong Show, from which she was fired for flashing her breasts. She also appeared on Rhyme and Reason and Match Game and in the 1980 "behind-the-scenes" movie version of The Gong Show. She also appeared on the Playboy Channel game show Everything Goes, and with her former Gong partner Jamie Farr on Hollywood Squares Game Show Week II in 2004.
Morgan appeared in the 2002 George Clooney–directed film Confessions of a Dangerous Mind, a fictionalized version of the life of Chuck Barris, creator of TV game shows The Gong Show, The Dating Game, and The Newlywed Game.