In a 40-year stage, film, and television career, Weston played sleazy villains to hapless men going through midlife crises to clumsy comics. He was one of the bad guys who victimize a blind Audrey Hepburn in the 1967 cult classic Wait Until Dark (1967) but meets his end at the hands of the number one Bad Guy Alan Arkin. He appeared in much lighter fare in Cactus Flower (1969), Please Don't Eat the Daisies (1960), and Dirty Dancing (1987). In the last of these, he ran the Catskills resort that set the stage for romance between Patrick Swayze and Jennifer Grey.

His stage work on Broadway included "California Suite" (1976) and "The Floating Lightbulb" (1971). For the latter, he received a Tony Award nomination for his role as the trashy manager. Also in 1981, Weston appeared opposite Alan Alda in The Four Seasons (1984) about the friendship between three middle-class, middle-aged couples. Other film credits include Stage Struck (1958), The Cincinnati Kid (1965), The Thomas Crown Affair (1968), The Ritz (1976), and Ishtar (1987).