Alessandro Striggio the younger began life in the upper echelons of Mantuan court society, and rose from there. He received his musical genes (and possibly, training) from both parents; Alessandro Striggio the senior was a well-known madrigalist also serving the Gonzaga court, and he married the singer and lutenist Virginia Vagnoli. Alessandro the son makes his first documented appearance in history as a player in the splendid orchestra assembled for the 1589 Medici wedding festivities. However, the boy first went to law school in preparation for that more lucrative career. His twenty years' of political service to the Mantuan Dukes, indeed, elevated him to noble status. In turn, Rossi served as secretary to the Duke, Mantuan ambassador to Milan, and Chancellor (by 1628). He eventually also attained the noble rank of Marquis. Neither rank nor position could save him, however, from contracting the plague; he died of the dread malady in Venice in 1630, while on a diplomatic mission.
Alessandro also maintained his early interest in music throughout his life. As early as the mid-1590s he was contracting with publishers to print the final posthumous collections of his father's madrigals. Striggio's place in music history was then sealed by his collaborations with Claudio Monteverdi in Mantua and afterwards. Striggio was a member of the Mantuan organization the Accademia degli Invaghiti, which comissioned Monteverdi to write his first opera, Orfeo (1607). Striggio himself took the example of Ottaviano Rinuccini's Euridice and composed the libretto for Monteverdi. Similarly, he collaborated with Marco da Gagliano for two opera libretti in 1608. Even after Monteverdi left Mantua to serve as Maestro di Cappella for San Marco in Venice, Striggio continued corresponding and collaborating with him. Striggio probably wrote at least two further libretti for Monteverdi, those for the (lost) cantata Apollo, and for Monteverdi's 1616 ballet Tirsi e Clori.