Before it settled into its present ecclesiastical sense, exarch was a secular administrative title: the Byzantine Emperor Maurice established the Exarchate of Ravenna and the Exarchate of Africa in the late sixth century as consolidated military and civil governorships covering Italian and North African imperial territory under sustained external pressure, each exarch holding combined civil and military authority considerably broader than an ordinary provincial governor's, a Byzantine innovation born of crisis rather than routine administration.
The ecclesiastical use of exarch as a delegated church authority developed alongside and partly in imitation of this secular office, borrowing the same basic idea, a delegate holding real authority over a defined territory on behalf of a higher power based elsewhere, and applying it to church governance instead of imperial government; the secular Exarchate of Ravenna itself fell to the Lombards in 751, ending the political institution centuries before the ecclesiastical title that borrowed its name and structure had run anywhere near its own course through the later history of the Eastern churches.
The title survives in ordinary present-day use through the apostolic exarchate, a canonical structure the Catholic Church uses for Eastern Catholic communities living outside their traditional homeland, such as Ukrainian or Syro-Malabar Catholics settled in Western Europe or North America, where a bishop titled exarch governs the diaspora faithful of that particular church much as an ordinary bishop governs a diocese, a modern administrative application of the same underlying idea of delegated authority over a defined body of the faithful that the historical rank always expressed.