Carney spent several years performing in vaudeville, developing his comedic timing and stage presence. His film career began in 1941 with a role in Convoy. Following this, he signed a contract with RKO Radio Pictures, where he gained recognition in supporting roles in films such as Mr. Lucky.

In 1943, Carney was paired with Wally Brown as RKO's answer to Abbott and Costello. The duo starred in a series of low-budget comedy films and also appeared together in Step Lively (1944), a musical remake of the Marx Brothers' Room Service. During this time, Carney and Brown also participated in a live USO tour organized by the studio. However, their partnership ended in 1946 after the release of Genius at Work, when RKO terminated their contracts.

After his split from Brown, Carney continued working in film and television as a character actor. One of his notable later roles was as Mayor Dawgmeat in the 1959 musical film Li’l Abner. He also appeared in television, playing Harry Nolan in the episode The Five Books of Owen Deaver (1958) from Have Gun – Will Travel. Carney reunited with Wally Brown in the films Who Was That Lady? (1960) and Walt Disney’s The Absent-Minded Professor (1961), though they did not share any scenes. The duo was supposed to reunite for It’s a Mad, Mad, Mad, Mad World (1963), but Brown passed away before filming began. During the 1960s and 1970s, Carney worked extensively with Walt Disney Productions, taking on numerous supporting roles. His final film appearance was in Herbie Rides Again (1974), released posthumously.

Alan Carney passed away on May 2, 1973, in Van Nuys, California, after suffering a heart attack. Reports suggest that his fatal heart attack was triggered by the excitement of winning the daily double bet at Hollywood Park Racetrack. He was 63 years old at the time of his death.