Fliegel made his debut with the Houston Symphony at 15, and joined the orchestra while still in high school. After serving in the air force during World War II, he became concertmaster, a position he held for 25 years, performing with such conductors as Thomas Beecham, John Barbirolli, Andr_ Previn, Lawrence Foster, Leonard Bernstein, Eugene Ormandy, and Christoph Eschenbach.
In 1972, after some health problems, Fliegel moved to the principal second violin position, where he remained until he retired in 1995. After retiring from performing, he remained on the faculty of the Shepherd School of Music at Rice University, and played with the Shepherd String Quartet.