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 much-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.