Watson grew up in Bonner Springs, Kansas, and Kansas City, Kansas. He attended the University of Miami along with fellow students Pat Metheny, Jaco Pastorius and Bruce Hornsby. After graduating in 1975, he moved to New York City and joined Art Blakey's Jazz Messengers. The Jazz Messengers, sometimes referred to as the "University of Blakey", served as the ultimate "postgraduate school" for ambitious young players. He performed with the Jazz Messengers from 1977 to 1981, eventually becoming the musical director for the group.
After completing his tenure as a Jazz Messenger, Watson became a sought after musician, working along the way with many notable musicians, including: drummers Max Roach and Louis Hayes, fellow saxophonists George Coleman and Branford Marsalis, multi-instrumentalist Sam Rivers and trumpeter Wynton Marsalis. In addition to working with a variety of instrumentalists, Watson has served in a supporting role for a number of distinguished and stylistically varied vocalists, including: Joe Williams, Dianne Reeves, Lou Rawls, Betty Carter, and Carmen Lundy, and has performed as a sideman with Carlos Santana, George Coleman, Rufus and Chaka Khan, Bob Belden and John Hicks.
Later, in association with bassist Curtis Lundy and drummer Victor Lewis, Watson started the first edition of Horizon, an acoustic quintet modeled after the Jazz Messengers but with its own slightly more modern twist. The group recorded several titles for the Blue Note and Columbia record labels.
In addition to his work as leader of Horizon, Watson also led a group known as the High court of Swing (a tribute to the music of Johnny Hodges), The Tailor-Made Big Band (16 pieces in all) and is a founding member of the 29th Street Saxophone Quartet, an all-horn, four-piece group with alto saxophonist Ed Jackson, tenor saxophonist Rich Rothenberg, and baritone saxophonist Jim Hartog. Watson also composed an original song for the soundtrack of Robert De Niro's A Bronx Tale (1993).
A resident of New York for most of his professional life, Watson served as a member of the adjunct faculty and taught saxophone privately at William Paterson University from 1985 to 1986 and the Manhattan School of Music from 1996 to 1999. He is currently involved with the Thelonious Monk Institute's yearly "Jazz in America" high school outreach program.
In 2000, he was approached to return to his native midwestern surroundings on the Kansas-Missouri border. Watson was selected as the first William D. and Mary Grant/Missouri, Distinguished Professorship in Jazz Studies. The past six years he has served as the director of jazz studies at the University of Missouri–Kansas City Conservatory of Music, although he still manages to balance concert engagements around the world with his teaching responsibilities. Watson's ensembles at UMKC have garnered several awards and national recognition.