Most Christians realize that when Jesus began what came to be known as the Lord's Supper, the context was the Passover Feast. By that point in Jewish history, the cup of the Passover was likely four cups of wine (the number found in the Mishnah) interspersed throughout the meal. This helps to explain why it is that in Luke's account there are actually two cups, one on either side of the bread.
In the Mishnah, the four cups are specifically called "wine." And what was the character of this drink? It was wine mixed with water:
When they have mixed the first cup of wine-- . . . . They mixed for him a second cup of wine. . . . They mixed the third cup for him (m. Pesahim 10:2-7).
The earliest specific references to these mixtures has a ratio of two parts water and one part wine (m. Niddah 2:7). Later, in the Talmud, the ratio is three to one (b. Erubim 54a and b. Nedarim 55a). The mixing of wine with water was evidently an early practice in Judaism as well as among the Greeks. For example, Song of Songs 7:2 speaks of "mingled" or "blended" wine, which is clearly thought of as being good. See for yourself.
When it came to "wine" or "the fruit of the vine," early Christian practice mirrored that of Judaism. For example, in that famous passage in Justin Martyr's First Apology which describes Christian worship, the elements of the Lord's Supper are bread and "wine mixed with water" (Apology I, 65). Similar passages are found in the writings of Hippolytus, Irenaeus, and Cyprian.
Someone might ask, But if the ancients used the term "wine" to refer to the wine-and-water mixture, then why does Justin mention both wine and water? Probably because he wanted to protect the church from the pagan accusation that Christians were drinking their wine straight, unmixed.
By way of summary, the wine of the ancient world and what we call "wine" are not the same things. Do both contain alcohol? Yes. But who today mixes wine with a larger amount of water? Or any water at all?
Were there people in the ancient world who got drunk on wine that was unmixed or only slightly diluted? Of course there were. But again, even the average pagan considered such practice to be pretty low. The only wine fit to drink had been mixed with a large amount of water, which was the universal practice.