It is believed the bassist had been suffering from a hip injury in recent weeks, which forced him to pull out of a number of shows across America.
Dusty, real name Joseph Michael, was best known as the group’s bass guitarist.
The musician was inducted into the Rock and Roll Hall of Fame as a ZZ Top member in 2004.
Away from the stage, he also starred in several TV shows, most notably playing himself in an episode of animated series King Of The Hill.