I'm not talking bards singing epics. I mean full on rehearse song and dance routines with a play as we would recognize them today.
If looking specifically for "book musicals" and are not counting operas and operettas (like those of Gilbert and Sullivan, who partnered up in the Victorian Era), according to Stanley Green's Broadway Musicals: Show By Show "American plays embellished with songs were being offered in New York as early as Colonial days, but 'The Black Crook' [Sept 12, 1866 for 475 performances, Book by Charles M. Barras, produced by William Wheatley and Henry Jarrett] was the first long running musical hit... Although the musical score was partly made up of popular numbers of the day and the grafting of songs, dances, and plot was crude, 'The Black Crook' claims an important place in theatrical history by introducing such staples of the musical-comedy form as elaborate scenic effects, colorful costumes, and rows of bare-limbed dancing girls." This is followed shortly by Rice and Goodwin's 'Evangeline', the works of Harrigan and Hart, Herbert and Blossom, and then at the turn of the century comes George M. Cohan. Early musical theatre were typically comedic or "lightweight" or simply musical revues like the Ziegfield Follies which debuted in 1907, especially once World War I started and people needed the distraction. It wasn't until Kern and Hammerstein's adaption of an Edna Ferber novel "Showboat" [December 27, 1927 for 572 performances] that the subject matter of musicals became heavier. According to Green "Both Jerome Kern and Oscar Hammerstein II had felt for some time that the Broadway musical theatre was suffering from too much sameness and tameness... Their efforts resulted in a recognized landmark in the history of theatre, one that broke ground in steering a course away from lightweight musical comedy and overweight operetta. Their characters were more three-dimensional, the music more skillfully integrated into the libretto, and their plot dared to deal with such unaccustomed subjects as unhappy marriages, miscegenation, and the hard life of black stevedores."