Rockwell was raised in Minneapolis, and in his early career he toured the U.S. in various rock and rhythm and blues bands. He worked in Las Vegas in the late 1960s and early 1970s, then moved to New York City, where he played with Thad Jones and Mel Lewis, Tito Puente, Ben Sidran, Freddie Hubbard, Ray Drummond, Billy Hart, Rufus Reid, Victor Lewis, Ron McClure, Tom Harrell, Chuck Israels, John Hicks, Al Foster, Anthony Cox, Bill Dobbins, Keith Copeland, Clint Houston, and Richie Beirach.
After settling in Copenhagen, Rockwell released a large number of albums on jazz label SteepleChase Records. He has also worked in Europe with Ernie Wilkins, Kenny Drew, Alex Riel, Marilyn Mazur, Kenny Wheeler, Jan Kasperson, and Jesper Lungaard.
- Androids (Celebration Records, 1974)
- No Rush (SteepleChase Records, 1985)
- On the Natch (SteepleChase, 1985)
- The Bob Rockwell Trio (SteepleChase, 1989)
- Reconstruction (SteepleChase, 1990)
- Trio (SteepleChase, 1990)
- Born to Be Blue (SteepleChase, 1994)
- Light Blue (SteepleChase, 1995)
- Shades of Blue (SteepleChase, 1996)
- After Hours (Go Jazz, 1999)
- It's All Right With Me (Pony Canyon, 2001)
- Love Eyes (Pony Canyon, 2001)
- Bob's Wilder: Music of Alec Wilder (Stunt Records, 2003)
- Bob's Ben: A Salute to Ben Webster (Stunt, 2005)