It's not Puritanism, but Judaeo-Christianity. In the Bible, God is always described as telling the truth, so any form of misrepresentation is moving yourself towards the opposite of God's character. Cheating is basically a form of misrepresenting yourself; commandment #9 prohibits false testimony about your neighbor, which presumably also includes false testimony about yourself. The New Testament also prohibits lying, as does the Talmud. In fact, the New Testament describes Satan as the father of lies, which presumably reflected Jewish traditions at the time. Imitating the devil would hardly be honorable for either Jews or Christians, and Paul even explicitly says we are to imitate Christ. (Since Paul was a rabbi applying the current Jewish thinking in the context of discovering that Messiah = Christ = God, presumably the Jewish thinking was that we should imitate God)

The Old Testament also offers an approach to dealing with rampant cheating in society: trust in God to provide you with what you need. It's not an exact match for classmates cheating, but David had to deal with a literal existential crisis by a king who believed falsehoods about him, and dealt with it by trusting God and refusing to kill the king (that is, using the king's methods to secure himself). Many of the psalms appear to record the emotional process he went through.