Byzantium was bound to survive longer than the western empire for a few reasons:
1. The East was much richer than the West, having been the cradle of civilisation for millennia. Rome alone could not equal the combined wealth of Alexandria, Antioch, the Greek cities and the rest of the Hellenised east. Whenever barbarians crossed the frontier, the eastern emperors used their gold to persuade them to go elsewhere - usually, they would head west.
2. Defendable core territory. While from the time of the Arab invasions until the reign of Basil II, most of the Balkans was overrun and out of imperial control, this mattered little in terms of Byzantium's survival. For one, the barbarians could never cross into Asia Minor, where some of the richest and most productive provinces lay. Moreover, the core of the empire was relatively impervious to invaders. This consisted not just of Anatolia, but also the Peloponnese, mountainous Thessaly, the well-defended cities of Macedonia - and, of course, Constantinople.
3. Constantinople is probably the chief reason for Byzantium's longevity. It's geographic advantages are almost miraculous. No city in the west had anything approaching these benefits, not even the fortified Ravenna. Many wonder why a single city bore so much importance when the vast majority of Roman subjects lived in villages and hamlets tending to their farms. Yet, in the end, all their surplus went to the cities, and in the east, all roads led to Constantinople - to a much greater extent than those of the west ever led to Rome.
Constantinople was the bridge between the European and Asian parts of the Empire. An emperor holding the city could safely lose all of the Balkans to his enemy yet still draw on the massive resources he had further east. Moreover, Constantinople was an important naval base and allowed its owner free access to the Pontic coast as well as the Aegean. To topple the Byzantines would beyond a doubt require the capture of Constantinople. And when Rome fell, the technology needed to do that was still nearly a thousand years away.