Picon was born in New York City, the daughter of Polish-Jewish immigrants: Clara (née Ostrow), a wardrobe mistress, and Louis Opiekun, a shirtmaker. Opiekun is a Polish language name meaning “guardian” or “caretaker”. Her surname was later changed to Picon. Her career began at the age of six in the Yiddish Theatre. In 1912, she debuted at the Arch Street Theatre in Philadelphia and became a star of the Yiddish Theater District, performing in plays in the District for seven years.

Picon appeared in many films, beginning with silent movies. Her early films were made in Europe; among the first was the Yiddish language East and West, filmed in Vienna in 1923, which is the earliest of her movies that survives. The film depicts a clash of New and Old World Jewish cultures. She plays a U.S.-born daughter who travels with her father back to Galicia in East Central Europe. Her husband Jacob Kalich played one of her close relatives.

Picon was so popular in the 1920s, many shows had her adopted name, Molly, in their title. In 1931, she opened the Molly Picon Theatre.

In 1934, Picon had a musical comedy radio show, the Molly Picon Program, broadcast on WMCA in New York City. In 1938, she starred another radio program on WMCA, I Give You My Life. That program “combined music and dramatic episodes that purported to be the story of her life.” Two years later, she starred in Molly Picon’s Parade, a variety show on WMCA.

Picon made her English language debut on stage in 1940. On Broadway, she starred in the Jerry Herman musical Milk and Honey in 1961. In 1966, she dropped out of the disastrous Chu Chem during previews in Philadelphia; the show closed before reaching Broadway.

Picon had a bit part in the 1948 film The Naked City as the woman running a news-stand and soda fountain towards the climax of the film. Her first major Anglophonic role in the movies was in the film version of Come Blow Your Horn (1963), with Frank Sinatra. One of her best-known film roles was as Yente the Matchmaker in the 1971 film adaptation of the Broadway hit Fiddler on the Roof.

Picon wrote a biography about her family called So Laugh a Little in 1962. In 1980, she published her autobiography, Hello, Molly!.

In the comedy For Pete’s Sake (1974), she appeared as an elderly madam (“Mrs. Cherry”) who arranges a disastrous stint for Barbra Streisand on a job as a call girl. She later had television roles on the soap opera Somerset and appeared in a few episodes of The Facts of Life as Natalie’s grandmother. Picon’s final role was as Roger Moore’s mother in cameo appearances in the comedies Cannonball Run & Cannonball Run II in 1981 and 1984, respectively.

Picon died on April 6, 1992, aged 94, from Alzheimer’s disease in Lancaster, Pennsylvania. Yankel Kalich, her husband from 1919 until his death in 1975, died from cancer. They had no children. She and her husband are interred in the Yiddish Theater section of the Mount Hebron Cemetery in New York City.