Apparently the Hurrian Hymn to Nikkal is the oldest complete written piece of music, but it seems it was rediscovered at some point, rather than being in continuous use.
I think there are some serious contenders to be found amongst Byzantine music, such as the Hymns of Saint Kassia (9th Century) and the so-called "Byzantine Kalanta" (c. 10th Century). These liturgical songs were sung throughout the later history of the Byzantine Empire, and remained popular even after the Empire's fall. They are still sung today in the Byzantine Orthodox liturgy, especially during the holidays.