Amsterdam was born Moritz Amsterdam in Chicago, Illinois, the youngest of the three sons of Max and Jennie (Finder) Amsterdam, Jewish immigrants from Austria-Hungary. He began working in vaudeville in 1922 as the straight man for his older brother's jokes. He was also a cellist, a skill he used throughout his career. By 1924, he was working in a speakeasy operated by Al Capone. After being caught in the middle of a gunfight, Amsterdam moved to California and worked writing jokes . His enormous repertoire and ability to come up with a joke on any subject earned him the nickname The Human Joke Machine. He sometimes performed with a mock machine on his chest, hanging by a strap. He turned a hand crank and paper rolled out; he would then read the machine's joke, although actually the paper was blank .

Amsterdam's reputation for humor preceded him. Hal Block tells of Amsterdam walking up Sixth Avenue in New York and meeting an old friend. "Where have you been?" the friend asked. "Sick," Amsterdam replied, "I've been in bed with a cold." His friend looked at him and asked, "What's so funny about that?"

During the 1930s, Amsterdam was a regular on The Al Pearce Show radio program, and by 1937 was the master of ceremonies on The Night Club of the Air.

Amsterdam also had a notable career as a songwriter, with his first popular success being "Why Oh Why Did I Ever Leave Wyoming." Amsterdam's most famous achievement as a songwriter was as the credited lyricist on the 1945 Andrews Sisters hit "Rum and Coca-Cola". However, the original version of the song (which Amsterdam did not write) was performed by a Trinidadian calypso singer named Lord Invader, and Amsterdam was subsequently involved in a copyright suit over the song, which dragged on until 1948. In the end, Lord Invader was given a substantial royalty payment for having written the original lyric to the piece, while Amsterdam retained the credit (and the publishing rights) for his revised version of the lyric.

In the early 1940s he was a screenwriter, contributing dialogue for two East Side Kids films.

By 1947, he was performing on three daily radio shows. Beginning in 1948, he appeared on the radio show Stop Me If You've Heard This One. The Morey Amsterdam Show aired on CBS radio from July 10, 1948 to February 15, 1949. For three months, it was on both radio and TV, using different scripts with the same premise and cast.

The Morey Amsterdam Show ran on CBS television from December 1948 to March 1949 and on DuMont from April 1949 to October 1950. Among Morey's regular guests was a song-and-dance man named Art Carney . The cigarette girl was future author Jacqueline Susann, wife of the producer of the show, Irving Mansfield . Jazz musician Johnny Guarneri led the band .

In 1950, he briefly hosted the comedy-variety show, Broadway Open House, television's first late-night entertainment show, on NBC. One of the pioneering TV creations of NBC president Pat Weaver, it demonstrated the potential for late-night programming and led to the later development of The Tonight Show. The show was originally to be hosted by comic Don "Creesh" Hornsby (so named because he yelled "Creesh" often), but he died of polio two weeks before the premiere broadcast. Hornsby's replacements, hosting different nights each week, were Amsterdam (Monday and Wednesday) and the raucous Jerry Lester (Tuesdays, Thursdays and Fridays), the brother of character actor Buddy Lester. However, Amsterdam soon exited the show, leaving Lester the sole host.

In 1957, he appeared as Jack Connors in the third episode entitled "The Pretenders" of the syndicated television sitcom, How to Marry a Millionaire, with Barbara Eden and Merry Anders .

His best-known role was as comedy writer Buddy Sorrell on The Dick Van Dyke Show (1961—66), a role suggested for him by his friend Rose Marie, who also appeared on the show. Amsterdam wrote lyrics for the show's theme song, which were never heard on the air but have been performed by Dick Van Dyke in concert .

Amsterdam was an occasional panelist on Match Game during the 1970s. He appeared as a small-time criminal in several episodes of the soap opera The Young and the Restless in the 1990s. Amsterdam and Rose Marie later appeared as panelists on The Hollywood Squares and guest-starred together in a February 1996 episode of the NBC sitcom Caroline in the City (his final television appearance).

He played Cappy, owner of the local nightclub, in the Beach Party movies of the 1960s. Amsterdam and Rose Marie also co-starred in the 1966 film, Don't Worry, We'll Think of a Title, a comedy co-written and co-produced by Amsterdam. The film features Richard Deacon, their co-star on The Dick Van Dyke Show, with a cameo by the show's co-producer Danny Thomas. In 1958 Morey appeared in the low-budget film Machine-Gun Kelly with Charles Bronson and did a notable dramatic turn in the 1960 noir classic "Murder, Inc." as Catskill nightclub owner Walter Sage, the first victim (according to the film) of the newly minted Murder, Incorporated.

Amsterdam died of a heart attack in Los Angeles, California on October 28, 1996 at age 87, and was survived by his wife Kay Patrick and their children, Gregory and Cathy. He was entombed at Forest Lawn - Hollywood Hills Cemetery in Los Angeles.