A Streetcar Named Desire premiered in Boston and Philadelphia, then in New York on December 4, 1947, to almost unanimously laudatory reviews. The New Yorker described Streetcar as "deeply disturbing—a brilliant, implacable play about the disintegration of a woman, or, if you like, of a society."

Streetcar was highly praised by its first director, Elia Kazan, who, from his knowledge of Williams' character, was one of the first to point out psychological similarities between Williams and Blanche. Kazan noted that "I keep linking Blanche and Tennessee ... Blanche is attracted by the man who is going to destroy her ... I also noticed that at the end of the play—all was an author's essential statement—Stella, having witnessed her sister's being destroyed by her husband, then taken away to an institution with her mind split, felt grief and remorse but not an enduring alienation from her husband ... The implication at the end of the play is that Stella will very soon return to Stanley's arms—and to his bed. That night, in fact. Indifference? Callousness? No. Fidelity to life. Williams' goal. We go on with life, he was saying, the best we can. People get hurt, but you can't get through life without hurting people."

Other critics were not always so appreciative or understanding. The distinguished American critic Mary McCarthy summarized Blanche with considerably less sympathy, remarking that in her character Williams had ''caught a flickering glimpse of the faded essence of the sister-in-law: thin, vapid, neurasthenic, romancing, genteel, pathetic ... a refined pushover and perennial and frigid spinster.'' McCarthy criticized Williams for crafting Blanche's character with the trappings of "inconceivable" tragedy and melodrama, commenting that the playwright's work "reeks of literary ambition as the apartment reeks of cheap perfume: it is impossible to witness one of Mr. Williams' plays without being aware of the pervading smell of careerism."

Audiences clearly disagreed: Streetcar ran for eight hundred and fifty performances on Broadway. It also won the Pulitzer Prize, the Drama Critics' Circle Award and the Donaldson Award.

The 1951 film adaptation won the New York Critics' Film Award and several Academy Awards.