Egri came to the US in 1906 and worked in a New York garment factory as a tailor and presser. He was an active member of the International Ladies' Garment Workers Union.

Egri wrote his first three act play at the age of ten, according to his biographical sketch in The Art of Dramatic Writing. In 1927, Rapid Transit, Egri's expressionist play, was translated from Hungarian and produced at the Provincetown Playhouse in New York. Casting about for some adequate means of conveying a sense of the furious pace of this machine age, Egri pictured a world in which all of life is compressed into twenty-four hours. Children grow to maturity in a few minutes; meals are eaten in split seconds; tabloid newspapers are issued at intervals of a second or two, and the loss of half a minute is a serious matter. The New York Times described the play as "chaotic at times, but sporadically interesting." Egri was also author of other plays, among them the satirical comedy Believe Me or Not (1933), Tornado (1938), This is Love (1945 with Arden Young) and The Cactus Club (1957). His one act Hungarian plays include Satan is Dead, Spiders, Between Two Gods, There Will be No Performance, and Devils.

Egri gave courses in playwriting, first in New York (1860 Broadway and 2 Columbus Circle), and then in Los Angeles. One student who garnered much attention was a 63 year old grandmother, Mrs. Esther Kaufman, who attended Egri's classes. Egri encouraged Mrs. Kaufman to write a play about growing up on the Lower East Side. The result was A Worm in the Horseradish, which had its premiere at the Maidman Theater in New York, March 13, 1961, and closed May 28 after mixed reviews. Egri also worked with other playwrights and screenwriters, including Woody Allen, who took Egri's course at 2 Columbus Circle. "I still think his The Art of Dramatic Writing is the most stimulating and best book on the subject ever written, and I have them all," Allen told biographer Eric Lax.

Originally published by Simon and Schuster in 1942 as How to Write a Play, Egri's treatise was revised and published as The Art of Dramatic Writing in 1946.

Egri argues in The Art of Dramatic Writing against Aristotle's view of character being secondary to plot (as stated in Aristotle's Poetics). According to Egri, well defined characters will create the plot for themselves, and so character is more important.

In 1965, Egri expanded on his views of character development and motivation in the book, The Art of Creative Writing.

Egri taught creative writing in his West Los Angeles home (at 11635 Mayfield Avenue) until shortly before his death. He died of a heart attack at Cedars of Lebanon Hospital.