When The Glass Menagerie reached the New York stage in 1945, it was a resounding success. A year earlier, it had also been successful in Chicago, despite poor weather which initially deterred the audience. According to Felicia Hardison Londre, writing in American Playwrights since 1945, "a crusade by the warmly enthusiastic Chicago critics" was launched to keep the play in production. It has remained popular, with staged as well as filmed versions appearing frequently, and it is considered to be one of Williams's most successful works. Indeed, writing in The Christian Century in 1964 while Williams was still alive, critic William R. Mueller stated that Williams "is the greatest living American playwright and ranks next to [Eugene] O'Neill in the history of American theater."

Critics almost inevitably remark on the poetic structure and language of The Glass Menagerie. As evidenced by the success with which his plays have been filmed, Williams brought a "cinematic concept of dramatic action to the American stage," according to Londre. She continued, describing Williams's work as characterized by "a harmonious blending and mutual reinforcement of dialogue, character, symbols, scenic environment, music, sound effects, and lighting." In his article Mueller stated that a "common denominator of Williams's plays is the quality of their poetry." Mueller defined this "poetry" not in terms of conventional poetic devices such as rhyme and meter, but as language "suffused with imagery and so phrased as to create a dreamlike state." In Tennessee Williams: A Tribute, S. Alan Chesler credited Williams with creating "a new poetic drama.... Williams has employed visual and auditory effects to previously unattempted extents by emphasizing color, music and scenic devices."

Yet poetry is far from the only characteristic for which critics have praised Williams and his plays. Although many of the stage directions in this play are almost novelistic in their detail, his work is also discussed in terms of its theatricality. Contrasting Williams with William Shakespeare, Mueller argued that "Shakespeare can be played without setting, lighting, costume, music; Williams cannot. He makes fullest use of the craft of the stage: scenic effects, lighting, color, music are of vast importance in evoking from the audience the desired emotional response." The use of a scrim between the audience and the actors at the beginning of the play would be one example of this. Another would be the frequency with which scene changes are signaled through fading music.

Critics also frequently comment on the psychological complexity of Williams's work, especially addressing the autobiographical roots of The Glass Menagerie. In part because of his success in creating characters who evoke empathy, even if they are not entirely typical, The Glass Menagerie and plays which soon followed appealed to an exceptionally broad audience, from high school students to professional critics. In the words of Foster Hirsch in A Portrait of the Artist: The Plays of Tennessee Williams, "Williams creates driven characters who are unlike anyone most of us are ever likely to meet and yet they are almost all convincing and recognizable." In an article published in Players, Gerald Berkowitz analyzed these characters in terms of the setting Williams has created for them: "as we discover each aberration or peculiarity in their [the Wingfields'] characters, we also discover that it is benign or even appropriate to their setting. Laura's pathological shyness does not stifle her at home; she is even able to overcome her fear of Jim when talking of her glass animals. Her lameness, which so embarrassed her in high school, becomes irrelevant when she is sitting in the apartment."

In addition to the number of awards Williams won during his lifetime, another way to measure his critical success, and the critical success of The Glass Menagerie, is through the professional attention he continues to receive. Books and articles continue to be written about this play as the thematic, literary, and theatrical issues it raises continue to be debated. Within the last generation, these publications include not only a wide range of American and Canadian periodicals but also journals published in Brussels, France, Brazil, The Netherlands, Germany, and South Africa. This play, in other words, has achieved not only significant popular success but international critical success.