In 1949, Weston appeared as Mr. Storm in episode 5 of Captain Video and His Video Rangers.

On television he made numerous appearances such as murderer Fred Calvert in the 1958 Perry Mason episode, “The Case of the Daring Decoy”. In 1961, he was a guest star in the TV drama Route 66, playing the manager of a traveling group of young women nightclub dancers, who mistreats his employees. In 1963, he was a guest star in an episode called “Fatso” in the TV drama The Fugitive.

Weston, a Cleveland, Ohio native, usually played comic roles in films such as Cactus Flower (1969) and Please Don’t Eat the Daisies (1960). He occasionally took on heavier parts, such as the scheming crook and stalker who, along with Alan Arkin and Richard Crenna, attempts to terrorize and rob a blind Audrey Hepburn in the 1967 film Wait Until Dark.

In 1976, he was nominated for a Golden Globe Award for Best Actor in a Motion Picture – Musical or Comedy for his performance in the film The Ritz. In 1981, Weston appeared on Broadway in Woody Allen’s comedy The Floating Light Bulb, for which he was nominated for a Tony Award as Best Actor. Other stage appearances included Bells are Ringing in 1956 (with Judy Holliday), The Ritz in 1975, Neil Simon’s California Suite (1976) and One Night Stand in 1980.

His second marriage was to Laurie Gilkes and they had one child together. They were married until his death from lymphoma on May 3, 1996, after a six-year struggle. He was 71 years old.