How were the Romans able to create and sustain a civilization over a period of almost ten centuries?

Many of the great empires of the ancient world, including the Egyptian, Chinese, and Persian empires, successfully lasted for many centuries.

One key feature in the success of an empire was an effective system of administration. The Romans borrowed many administrative ideas from the great empires of the east, especially the Persians. Important types of infrastructure which facilitated administration of a large empire were a massive road network, a system of couriers, and a sort of postal system that facilitated communication from the center to the periphery of the Empire. Another important element was letting local elites of conquered territories continue to conduct local level administration. 

Next, the Romans had a very powerful and well-organized army. Not only did they innovate in military technology, but they created a path to Roman citizenship for barbarians through service in the legions, something that meant that rather than becoming hotbeds of rebellion, young men from conquered territories had a clear path to success and land ownership. 

Finally, the Romans were relatively fair and just rulers of conquered territories. Although conquered peoples had to pay taxes and become part of the empire, they were allowed to keep their local traditions and religions, meaning there was little incentive to rebel.