Middle East was always more "civilized" (judging by development of civilization) than Europe, till I believe Renaissance when thanks to science, inventions and capital/first banking Europe got finally ahead.
Central Asiatic Nomads the Huns are a different story.
Well, I think you might get an argument in terms of the classical world. I'm pretty sure that while the Greeks and Romans certainly didn't view the inhabitants of Anatolia and the Levant the way they did the peoples north of the Black Sea, for example, they did consider themselves more "civilized" than the peoples of the " inner Asia" of that time, and I would tend to agree, especially as time went on.
Just in terms of this use of the word "Asia"... As Aberdeen has pointed out, the specific focus wasn't made clear by the OP.
I probably shouldn't be speaking for him, but I think the OP may be looking at this from the point of view of the Greco-Roman world.
To the classical Greeks and Romans, the only "continents" of which they were aware were Europe and "Asia". "Asia" was everything to the east of the Aegean and the Mediterranean in general. Anatolia, or the area between the Aegean, Mediterranean and Black Seas, was Asia Minor., whose eastern boundary was formed by the mountains of western Mesopotamia. This was the "Orient" to them. The term sort of hung on...Oriental studies isn't just the study of the far east; it's, or at least was, also the study of Anatolia, the Levant etc.
Asia Major was east of the Don and Black Sea, including Sarmatica Asiatica, Colchis, Iberia, Armenia, Syria, Arabia, Babylonia, Mesopotamia, Assyria, Media, Susiana, Persis, Ariana, Hyrcania, Margiana, Bactriana, Sogdiana, India, the land of the Sinae and Serica.
All of these areas were "the other" to them, or the lands of the "barbarians", and they were in conflict with them, although it's certainly true that they also borrowed from them, especially, of course, from the peoples of the ancient Near East, whom, as I pointed out above, they certainly viewed differently than the people of Sarmatica, for example.
In terms of influence, I think it could be said that after the Persians were turned back by Greece, the pendulum swung decidedly in the other direction, and the classical civilization of Greece prevailed in Asia Minor and the Levant, with Greek culture, albeit mixed with "Asian" influences, dominating most of the east.
The ascendancy of Rome replaced Greek rulers with Roman administrators. While the Roman West fell, the Roman-Greek East held on, and maintained control of the area until the invasions from what might be called, I suppose, Asia Major.
In terms of influence East-West from that point on, a lot of it consisted of classical era learning preserved and transmitted both by the Byzantines and the Muslim empires that co-existed with, and then largely replaced them.
As for the fall of Constantinople, I don't see how it can be viewed as anything but an unmitigated tragedy. There's an excellent, not too long book about it called The Fourth Crusade and the Sack of Constantinople by Jonathan Phillips. It's very enlightening.
It may seem a long time ago to some people, but the implications were huge. Just one consequence is the lasting suspicion on the part of the Eastern Orthodox Christians of the bona fides of the Latin Christian west.