In 1961, Elston joined veteran radio broadcaster Loel Passe to announce the final season of Houston's minor league franchise, the Houston Buffs. With the expansion of the major league and the inaugural 1962 season of the Houston Colt 45s, Elston was chosen to lead the radio broadcast. (Loel Passe stayed on as the color commentator with Elston until Passe retired in 1976.) The team changed its name to the Astros three years later, and Elston continued as their main announcer through 1986, when he ended his association with the Astros and joined Tal Smith Enterprises as a consultant and researcher.
Starting in 1987, Elston went back to calling national radio broadcasts instead of games for a specific team. He called the CBS RadioGame of the Week until 1995, and also called postseason NLDS games on CBS Radio in 1995, 1996, and 1997. He then retired from broadcasting.
In 2006, Elston was awarded the Ford C. Frick Award from the Baseball Hall of Fame. The award is given annually to a baseball announcer who has given major contributions to the game. Elston was healthy enough, at the age of 84, to accept the award in person at Cooperstown.