I was gonna say Carousel...I guess you could say it's the first very well-known musical to not have an overture. It's funny, I'm reading a book analyzing the music in the best musicals, and I just started Carousel, and read about the Carousel Waltz and why they decided to do that in place of an overture and why it worked. Rodgers was never a fan of overtures, he always felt that there was too much brass and not enough strings, so the only parts you could hear are melodies that are blasted. There's this whole debate about whether the Carousel Waltz was originally written for this radio program, but anyway...it introduces all the major musical elements that makes the music to Carousel so distinct.
And let's not forget West Side Story in terms of shows without overtures long before Chorus Line. (Even though the production that's currently licensed does include one, the original just started with the Prologue. Bernstein wrote the overture but then decided not to use it).
"If there was a Mount Rushmore for Broadway scores, "West Side Story" would be front and center. It snaps, it crackles it pops! It surges with a roar, its energy and sheer life undiminished by the years" - NYPost reviewer Elisabeth Vincentelli