Elliot was a member of the funk band Tower of Power, playing tenor sax for five years during the 1980s. He also worked on Stacy Lattisaw's 1986 album Take Me All the Way. His solo career took off when he remade the Percy Sledge classic "When a Man Loves a Woman", which had also been remade by Michael Bolton. Other well-known songs include "In the Groove," "Take Your Time," "Crush," "Chill Factor," "Corner Pocket," and "Sly," as well as remakes of classic pop songs such as "I'm Not in Love" and the Luther Vandross hits "Here and Now" and "Your Secret Love."
Elliot participated with fellow saxophonists Paul Taylor and Gerald Albright, as well as keyboardist Jeff Lorber, in the "Grooving for Grover" concert series. The series honored the legacy of the late Grover Washington, Jr.
Elliot was known for his zebra-striped tenor saxophone. In 2011, he switched from LA SAX Saxes to Dakota Saxophones, which the owner, Peter LaPlaca, sold for Dakota.