Over his career, Azcue threw out 45.17% of the base runners who tried steal a base on him, ranking him 10th on the all-time list. During the 1966 season, he threw out 62% of the base runners who tried steal a base, the fifth highest season percentage in major league history. He caught two no hitters in his career, Sonny Siebert in 1966 and Clyde Wright in 1970.
In an eleven-year major league career, Azcue played in 909 games, accumulating 712 hits in 2,828 at bats for a .252 career batting average along with 50 home runs, 304 runs batted in and an on-base percentage of .304. A capable defensive catcher, Azcue led American League catchers in fielding percentage in 1967 and 1968. His .992 career fielding percentage was second only to Elston Howard among major league catchers at the time of his retirement.