Harry Edison was born on October 10, 1915, in Columbus, Ohio, and spent his early childhood in Louisville, Kentucky, where an uncle introduced him to music. He returned to Columbus at twelve and began playing trumpet with local bands, including a group led by Earl Hood that gave him his first paid work while he was still in high school. In 1933 he joined the Jeter-Pillars Orchestra, then worked with the Mills Blue Rhythm Band and Lucky Millinder's orchestra before moving to New York in 1937 and joining the Count Basie Orchestra that same year.

The nickname that became his professional name came from that band. As Edison told DownBeat in 1956, he was the youngest man in the group in March 1937, and one snowy day, while the musicians waited in the lobby of New York's Woodside Hotel for the bus to a run of one-night stands, Lester Young teased him about his sweet style and began calling him "Sweetie Pie." The name was shortened to "Sweets" within a few months and stayed with him for the rest of his life, which is why nearly every record and review bills him as "Sweets" while this profile is filed under Harry Edison. In the Basie trumpet section he worked beside Buck Clayton, and his section mates included Young, Buddy Tate, Freddie Green, and Jo Jones. He came to prominence as a soloist with the band and wrote and arranged for it occasionally, and he appeared with Young in the 1944 short film Jammin' the Blues, directed by Gjon Mili. Edison spent thirteen years with Basie, until the orchestra was temporarily disbanded in 1950.

Afterward he led his own groups, toured with Norman Granz's Jazz at the Philharmonic, and freelanced with other orchestras before settling on the West Coast in the early 1950s, where he became one of Los Angeles's most sought-after studio trumpeters. He is heard on records by Billie Holiday, Frank Sinatra, Nat King Cole, Billy Daniels, Margaret Whiting, Bing Crosby, and Ella Fitzgerald. His work for Sinatra and the arranger Nelson Riddle is the best known of this period: Riddle gave him a microphone separate from the rest of the trumpet section, and he answered Sinatra's vocals with short phrases played through a Harmon mute, a role the National Endowment for the Arts singled out on Songs for Swingin' Lovers! (1956). In 1955 he made two quintet albums for Granz's Norgran label, Buddy and Sweets with Buddy Rich and Pres and Sweets with Young. In 1956 he began a run of three albums with Ben Webster, including Gee Baby, Ain't I Good to You (recorded 1957), and in January 1957 he was part of the small group behind Holiday on the Verve sessions that produced Songs for Distingué Lovers. Leader dates followed for Verve and Roulette, among them Mr. Swing, Sweetenings, and Patented by Edison (1960), a quintet with Tommy Flanagan, Elvin Jones, the tenor saxophonist Jimmy Forrest, and the bassist Tommy Potter, and Together (1961), a collaboration with the singer Joe Williams.

Through the 1960s and 1970s Edison worked in studio orchestras for television, including The Hollywood Palace and The Leslie Uggams Show, played on Sinatra specials, and was prominent on the soundtrack and soundtrack album of the 1972 film Lady Sings the Blues. From 1973 he was musical director for the comedian Redd Foxx at theatre dates, concerts, and in Las Vegas. Back with Granz, now at Pablo Records, he recorded The Trumpet Kings Meet Joe Turner (September 1974) with Dizzy Gillespie, Roy Eldridge, and Clark Terry, a duo album with Oscar Peterson (December 1974), and Edison's Lights (1976), on which Basie himself plays piano on four tracks, followed by a series of albums with the tenor saxophonist Eddie "Lockjaw" Davis. He appeared frequently in Europe and Japan until shortly before his death, was the Los Angeles Jazz Society's first Tribute Honoree, and was named a Jazz Master by the National Endowment for the Arts in 1992.

Edison passed away at his home in Columbus, Ohio, on July 27, 1999, of prostate cancer, at the age of 83.

Edison's sound was built on economy. As the writer Will Friedwald has noted, he was a master at saying a great deal in very few notes, and many of his parts on Sinatra records amount to a handful of well-placed phrases from a Harmon-muted horn. The approach came out of the Basie band, where a relaxed beat and plenty of space did as much work as the notes, and his tone was the warm, singing one that earned him the nickname. Singers and leaders returned to him repeatedly: Lester Young and Billie Holiday had worked beside him in the 1930s Basie band and sought him out again for Granz's 1950s sessions, a pattern repeated with Ben Webster, Buddy Rich, Oscar Peterson, and Eddie "Lockjaw" Davis, who became a regular partner late in his career. In the Pablo "trumpet kings" sessions he stood beside Roy Eldridge, Dizzy Gillespie, and Clark Terry, setting his understatement against louder, more virtuosic styles. His most lasting contribution was as a Hollywood studio musician, whose muted trumpet backs singers on many records, and he kept performing in the United States, Europe, and Japan until shortly before his death.