To clarify... the Roman Empire encompassed the entire Mediterranean at it's peak, and most of northern Europe to boot. Their influence was so pervasive for so long that no less than three modern languages outside of Italy are deeply rooted in Latin.
And they controlled Greece. Not only did the Romans control Greece, they turned the city of Byzantium into a second capital - Constantinople! Further, this part of the Empire far outlived the Rome of Italy, remaining the seat of the Roman emperor for a thousand years after the Western Empire collapsed.
And yet... the Greek language remained Greek.
Why?
There are a few reasons:
The Romans looked up to the Greeks, revered their classics, respected their philosophy, etc. They did not desire to eliminate the Greek culture and the educated Romans voluntarily learned Greek as part of their education.
Also, Greek was very widespread as an administrative language throughout the East. Anatolia, Syria, Palestine, Egypt were all important provinces where Greek was widespread. Supplanting that with Latin would have been challenging, but not impossible. Many languages throughout history have supplanted their predecessors to become the Lingua Franca in that region (Akkadian, Aramaic, Greek, Arabic).
Contrast this with Gaul where the indigenous population was considered to be "lesser" than the Romans as well as hostile. Romanizing the Gauls was an essential component of the long-term pacification of the region.
We're getting lots of questions pertaining to the Greek Language this week!
Greek was widely spoken in the entire Eastern Mediterranean, and Greece was the focal point of Mediterranean culture. The Romans themselves were more Greek-ized than the Greeks were Roman-ized. There's more information in the thread I linked.