Robert Lrod Dorough was born on December 12, 1923, in Cherry Hill, Polk County, Arkansas, and grew up in Plainview, Texas. He came to jazz through records by Benny Goodman and Harry James. During World War II he served in Army bands as a pianist, clarinetist, saxophonist and arranger, and afterward he studied composition and piano at North Texas State University, graduating in 1949. From 1949 to 1952 he was a graduate student at Columbia University in New York and played piano in local jazz clubs on the side.

In the early 1950s he began to sing as well as play. He was hired for a tour by the boxer Sugar Ray Robinson, who had stepped away from the ring to try a career as an entertainer, and from 1954 to 1955 he lived in Paris, working as a musician and musical director. On October 28, 1954, he recorded there as pianist with the bassist Buddy Banks, the guitarist Jimmy Gourley and the drummer Roy Haynes. He also sang in the vocal group the Blue Stars with Blossom Dearie and Christiane Legrand, and he recorded with Dearie. Back in the United States he settled in Los Angeles, played clubs there, and for a time worked between the sets of the comedian Lenny Bruce. He had signed with Bethlehem, and his first album, Devil May Care, was released in October 1956, with the bassist Bill Takas, the trumpeter Warren Fitzgerald, the vibraphonist Jack Hitchcock and the drummer Jerry Segal. It included a version of Charlie Parker's "Yardbird Suite" with lyrics he had added.

Miles Davis liked the album. In 1962, when Columbia asked Davis for a Christmas single, he sought out Dorough, who wrote the lyric and sang "Blue Xmas (To Whom It May Concern)," which appeared that October on the anthology Jingle Bell Jazz. At the same session Dorough recorded "Nothing Like You" with an orchestra arranged by Gil Evans. It was the first time Wayne Shorter recorded with Davis, and it was released only in 1967, as the closing track of Sorcerer, which makes Dorough one of the few singers with a vocal on a Miles Davis record. In that same year, 1962, he put a lyric to Ben Tucker's blues "Comin' Home Baby." Mel Tormé recorded it in New York on September 13, 1962, and the single reached number 36 on the Billboard Hot 100 and drew two Grammy nominations. Dorough's second album, Just About Everything (Focus), followed in 1966, and in 1964 he released an RCA Victor single as the duo Bob and Al with the guitarist Al Schackman, who also played on his Focus album.

In the late 1960s he worked as a producer and arranger, making two albums with the producer and guitarist Stuart Scharf for the folk-pop group Spanky and Our Gang, and he was arranger, choir singer and pianist on Allen Ginsberg's setting of William Blake, Songs of Innocence and Experience (1970). Meanwhile an advertising executive, David McCall, asked him to put the multiplication tables to music. The result, "Three Is a Magic Number," led to Multiplication Rock, which began airing on ABC's Saturday morning schedule in January 1973 as the first season of Schoolhouse Rock! Dorough was the series' musical director, wrote and sang most of that season's songs, and later wrote "Conjunction Junction" (first aired November 17, 1973), which was recorded in Los Angeles with a jazz band: Jack Sheldon on trumpet and vocal, Teddy Edwards, Leroy Vinnegar on bass, Nick Ceroli on drums, Dave Frishberg and Dorough on piano, and Scharf on guitar. He wrote and directed episodes through 1996, and again for video releases in 2002 and 2009. The Multiplication Rock soundtrack brought a Grammy nomination for Best Recording for Children in 1974, and De La Soul sampled "Three Is a Magic Number" in 1989.

Through all of this he kept making jazz records, mostly on small labels: play-along albums for Music Minus One in the 1970s, a run of duo albums with the bassist Bill Takas from 1976 to 1986, Hoagy's Children (1983) with Dick Sudhalter and Barbara Lea, and Skabadabba, Songs of Love (with Art Farmer on flugelhorn) and This Is a Recording of Pop Art Songs in the late 1980s and early 1990s. From 1985 to 1993 he toured Europe several times with the saxophonist Michael Hornstein, Takas and the drummer Fred Braceful. At 73 he signed his first contract with a major label, Blue Note, which issued Right on My Way Home in 1997, then Too Much Coffee Man, with Phil Woods and Billy Hart among the players, and the live Dave Frishberg duet album Who's on First? in 2000. A Sunday brunch residency at the Iridium in New York produced the live Sunday at Iridium (Arbors, 2004), and a tour of England in his eighties ended with the recording of Small Day Tomorrow (Candid, 2006). He mentored the singer Nellie McKay, whom he met in 1997, and appeared on her albums Obligatory Villagers (2007) and Normal as Blueberry Pie (2009). His last albums were But for Now (Enja, 2015), a trio set with Hornstein and the bassist Tony Marino, and Live at the Deer Head Inn (Deer Head, 2016), recorded in Delaware Water Gap, Pennsylvania, on December 12, 2015, his 92nd birthday.

Dorough passed away at his home in Mount Bethel, Pennsylvania, on April 23, 2018, at the age of 94, and his wife, Sally Shanley Dorough, confirmed his death. He had been inducted into the Arkansas Jazz Hall of Fame in 1998, received a Pennsylvania Governor's Award for the Arts in 2002, and been awarded an honorary doctorate by East Stroudsburg University in 2007. The National Endowment for the Arts named him a 2019 NEA Jazz Master in an announcement made in July 2018, and in 2019 the Schoolhouse Rock! soundtrack was chosen for the Library of Congress's National Recording Registry.

Dorough was a pianist and singer of the bebop and cool schools whose real specialties were the lyric and the vocalese line: he wrote words for jazz instrumentals ("Yardbird Suite," "Comin' Home Baby") as well as for his own tunes, and he delivered them in a dry, conversational, quick-witted voice that never sounded like a conventional crooner's. AllMusic's biographer called him an adventurous master of vocalese and scat singing who, directly or indirectly, influenced Mark Murphy, Michael Franks, Mose Allison and Kurt Elling. The National Endowment for the Arts described a jazz career of more than seventy years that was well known in the jazz world before Schoolhouse Rock! made his songs familiar to millions of children. That television work was itself made with jazz players and jazz harmony: Jack Sheldon, who sang and played trumpet on "Conjunction Junction," said the band was a jazz band and that "rock" was only the label on the series. His reputation within jazz stayed modest in part because his steady income came from educational songs and his records appeared on small labels, but the Royal Bopsters, the vocal group led by Amy London, asked him to guest on their first album alongside Jon Hendricks, Annie Ross, Mark Murphy and Sheila Jordan.