It is best to think of of the Byzantine Empire as Christianized Rome. We tend to think of ancient empires as very separate, which distinct timelines and locations, but in reality, the lines are a lot more blurry. The Byzantine Empire was a direct continuation of Rome. The transition from Rome to Byzantium was not a takeover or an invasion, but the direct result of the Christianization of Rome which began around 30 C.E. The exact dates for the "fall of Rome" are debated, but generally historians see Rome's decline as taking place in the 300s, about the same time that we now consider to be the rise of the Byzantine empire. The rise of the Byzantine Empire is considered analogous to the rise of Christianity because its first emperor, Constantine I, famously converted to Christianity after claiming to see the symbol of Christ, the Chi Ro, on the battlefield. Byzantium can be considered a continuation of Rome in that it was a transition empire from pagan Rome to medieval Christian Europe.
