Depends on the specific empire in question, but for the most part, yes, they did. To use Rome as an example, they instituted Roman infrastructure, civil service, and government in each of their provinces, each of which paid tax and supplied soldiers to Rome.

Roman law applied in said provinces, and eventually the natives were, to varying degrees, 'romanized.' Many took on the language and traditions of their conquerors, to the point where, given a few centuries, Provincials came to rule the entirety of The Empire, some of whom are still remembered as among the greatest leaders Rome ever had.

They managed this despite the massive distance between Rome itself and the provinces by devolving authority to regional government. Rome still held supreme authority, but the provinces were theoretically capable of governing themselves, as they possessed governmental structures of their own, their own soldiers, and their own bureaucracy.

But in short, yes, some ancient empires actually Ruled that much land. No, they didn't just march their armies in and proclaim victory before getting out of there - when Rome occupied a place, it was eventually romanized to some degree, infrastructure was built, armies were raised from its citizenry, the reward for serving as such being Roman Citizenship for a Provincial, and above all, taxes were paid. Everyone knew who ruled them - his face was on their coins.

This is an interesting question, and I have a related one: In imperial Rome for example, was the power and influence of the Roman government as evident in the frontiers as it was in Rome itself?

In the case of the Assyrian Empire, yes, the subjects did in fact know who was ruling them. However, despite the taxes and tributes, the separate villages were for the most part left alone, unless a particular one had some worth (trade, culture, etc.). Obviously, there can't be a platoon stationed every quarter mile or so to guard all the land, so land deemed unimportant, while still 'belonging' to the Assyrians, was left largely unguarded. Those towns I mentioned above as having worth in some way, be it trade or treaty, were protected, as were the routes leading to them (again for trade purposes). So while not all the land was really 'claimed' besides some forms of tribute the Assyrians required, it was still considered part of the empire. Thats why smaller 'kingdoms' were able to sprout up within the empire, since it wasn't all monitored all the time. However, those rebellion 'kingdoms' were usually put down quickly (except in a few cases), as part of the kings' duty to the land and to prove himself worthy required him to go on long campaigns yearly. One tactic that needs to be mentioned is that the Assyrian King (and many other empires' kings employed this as well) took part in forced relocation. Commoners were less likely to rebel in a land that they didn't know about rather than their own land they've inhabited for generations.