Picon was born as Małka Opiekun in New York City, the daughter of Polish Jewish emigrants: 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 years 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 was so popular in the 1920s that many shows had her adopted name, Molly, in their title. In 1931, she opened the Molly Picon Theatre.

She appeared in many films, starting with silent movies. Her earliest films were made in Europe; among the first was the Yiddish film East and West, made in Vienna in 1923, which is the earliest of her films 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.

In 1934, Picon had a musical comedy radio show, the Molly Picon Program, on WMCA in New York City. In 1938, Picon starred in I Give You My Life on the same station. 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 (also on W.M.C.A.).

Picon's most famous film, Yidl Mitn Fidl (1936), was made on location in Poland and shows her wearing male clothing through most of the film. In the film, a girl and her father are forced by poverty to set out on the road as traveling Musicians. For her safety, she disguises herself as a boy, which becomes inconvenient when she falls in love with one of the other Musicians in the troupe. Later Mamele was made in Poland.

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 quit the disastrous Chu Chem during previews in Philadelphia; the show closed before reaching Broadway.

She 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. She portrayed Yente the Matchmaker in the 1971 film adaptation of the Broadway hit Fiddler on the Roof.

On an ironic note, in 1959 she was featured on an episode of the NBC-TV series Startime. It was an adaptation of Samson Raphaelson's play The Jazz Singer starring Jerry Lewis, in which she played Lewis's mother, Sarah Rabinowitz. In one scene, Lewis says the line, referring to Picon as his mother, "She's still in our presence, ladies and gentlemen, the Matchmaker".

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

In the comedy For Pete's Sake (1974), she played a madam ("Mrs. Cherry") who arranges a disastrous stint for Barbra Streisand on a job as a call girl. She later played a role on television on the soap opera Somerset and appeared in a couple of episodes of The Facts of Life as Natalie's grandmother. Her final roles were cameo appearances in the comedies Cannonball Run & Cannonball Run II as Roger Moore's mother.

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. Also buried there is Ida Kaminska, who like Picon, operated her own Yiddish theatre.