Christianity's relationship with Plato is so great, that Nietzsche even made the remark that Christianity is a "popular platonism", or platonism for "the many". Before Nietzsche, in the 'Renaissance' many scholars in the West asked the Pope to name Plato a Saint! The demand may be inappropriate, but it shows how impressive this relationship is. Before that, already in the beginnings of Christianity, some Christian thinkers were puzzled by this relationship. Clement of Alexandria wrote that "Greek philosophy is a clear picture of the truth, a divine gift that was delivered to the Greeks... Before the advent of the Lord, philosophy was necessary to the Greeks for righteousness. And now it becomes conducive to piety; being a kind of preparatory training to those who attain to faith through demonstration... Perchance, too, philosophy was given to the Greeks directly and primarily, till the Lord should call the Greeks..." Augustine, Gregory of Nyssa, Origen... - there exists a long series of Christian thinkers and Fathers that demonstrates how strong this impression is.
Some explain the relationship with the fact that Christian faith was adopted by Greeks and through them was developed and spread to the rest of the world. (Cf. Greek Orthodoxy - From Apostolic Times to the Present Day). Whatever the explanation, Christianity is certainly very close to Plato.
The expression "to control our flesh", although not far from truth, is a little problematic, because, under certain conditions, it might also apply to efforts towards a "super-man" condition - the way, e.g., buddhism or induism suggest.
In Christianity everything is a gift of God, the very fact that we are able to talk, to have a language, to make this conversation here and now, to fast or pray, - everything, the small and the great. The only thing that is ours, is the will to do this and not that, but all the powers that realize our will, are granted by God.
Therefore, if we were to talk about fast in Christian terms, we wouldn't say that it is a means to control our flesh, but that it is one among many gifts God gives us in order to help us get closer to Him. Even in Plato, the aim is not to "control our flesh" but to set our soul free from desires that hinder the knowledge of God. Eastern, Orthodox, Christianity, although keeping fast days, does not place importance on fasting as such.
Some fathers warn, that most important is the Fast that God himself sends. E.g. it is easy not to eat very much some days, but it is difficult to accept as a gift the loss of your property, poverty, illness..., - anything that God decides not to give or to take away, is a fast much more important that the fast we can decide.
Therefore, fast in our tradition acquired a new sense, not so much the sense of setting us free from our desires, as the sense of honoring God. In Easter, for example, we don't fast in order to avoid bodily passions, but because we can not have fun and enjoy while God is on the Cross.