Answer

Hello N.,
There is a whole series of laws relating to the authority of the sages that have no source. They uproot a Torah law through passive non-action. And some of the medieval authorities discuss active uprooting as well (for example, executing an informer and the like). There is punishment not according to the formal law (which is of course one aspect of this general authority). And of course there is the power of a religious court to confiscate property, and “whoever betroths, betroths subject to the consent of the rabbis,” and so on. According to Nachmanides, all of their authority to legislate (enactments and decrees) is of that kind.
It is not reasonable to hang this on the mechanism of a positive commandment overriding a prohibition, because there is not always a positive commandment, and not every prohibition is overridden by a positive commandment, etc. According to Nachmanides, it is not relevant at all, because there is neither a positive commandment nor a prohibition. So it is clear to me that this is not some kind of catch-all mechanism of that type. There is something else going on here.
As for some of these cases, I have long thought that this is an authority the sages took for themselves because there was no alternative. And even when a source appears, it is a post facto justification. The power of a religious court to confiscate property, and “whoever betroths…,” in my opinion, is a legal authority based on Rabbi Shimon Shkop’s conception of the law of civil matters. The claim is that personal status and the laws of acquisition and ownership are areas that every legal system sees as both its right and its duty to regulate, and therefore in Jewish law too the legal sphere (civil law and some of family law) operates in that way. Without public recognition of ownership or partnership/marriage (personal status), these things have no validity. You do not need a source for this, because it is a legal conception that is true for all people in the world, and for us as well. The sages’ consent to ownership and betrothal expresses a public stamp of validation, without which these acts have no legal force. I would argue even more than that: that the sages can dissolve a betrothal in the middle of the marriage, not retroactively — what scholars call “prospective dissolution” (Avishalom Westreich and others have noted this and brought sources for it from halakhic decisors). And the reason is that what ceases is not the act of betrothal, but the recognition of the couple as a married couple. And where there is no recognition, there is no marriage/betrothal.
What about punishment not according to the formal law, and uprooting a Torah law? Some time ago I saw a law in Maimonides that shed light on the matter for me. Laws of Rebels 2:4:
 

And the court has the power to uproot even these matters temporarily, even though it is inferior to the earlier courts, so that these decrees should not be more stringent than the words of the Torah itself; for even words of the Torah, any court may uproot as a temporary ruling. How so? A court that saw a need to strengthen the religion and make a fence so that the people should not violate matters of Torah may flog and punish not according to the law, but they may not establish the matter for generations and say that this is the law. And likewise, if they saw a temporary need to suspend a positive commandment or to transgress a prohibition in order to bring many back to the faith, or to save many Jews from stumbling in other matters, they do according to what the time requires. Just as a physician cuts off this one’s hand or foot so that the whole body may live, so too the court may instruct, at certain times, to violate some commandments temporarily so that all of them may endure, in the way that the early sages said: desecrate one Sabbath for him so that he may keep many Sabbaths.

 
First, the argument he makes from rabbinic law to Torah law is puzzling, because in the first two laws of that same chapter he rules that uprooting Torah law is easier (for which a different count is sufficient) than uprooting rabbinic law (which also requires superiority in wisdom and number over the previous court). It seems clear that here he is speaking about freezing the law, not changing it. I am not disagreeing with earlier generations and changing the law they established; rather, I am freezing it. Here there is no need for the requirements of authority (number, superiority, etc.), because this is not a dispute. What is required here is a recognized religious court in the relevant generation that has authority to decide for its generation. If it reaches the conclusion that a certain law is harmful (not mistaken — because then this would be a dispute with the earlier authorities), it has the right and the duty to freeze that law temporarily. And, as is well known, nothing is more permanent than the temporary, so such freezes remain in place until the coming of the righteous redeemer, or until sages arise in some generation when matters become unbearable and they freeze the suspension itself (or thaw it out).
Notice that Maimonides gives as his example here the authority to punish not according to the formal law. In other words, this freezing is temporary, but precisely because of that, the authority for it is entrusted to every court in every generation. You do not need ordained judges, and not a Sanhedrin, and nothing at all (no requirement is mentioned here, unlike the previous laws there). Clearly, this is an authority based on reason, because this is not a halakhic determination, and therefore no source is required. Perhaps one could say that the public accepted the Torah on the assumption that it would have the ability to maneuver with it so that people could live by it and observe it in a sensible way.
Now I return to an old thesis of mine regarding punishment not according to the formal law, and extend it to all these enactments of uprooting and freezing. I argue that originally this authority was given to the king, because this is not an internal halakhic authority but one belonging to an institution parallel to Jewish law and to the Sanhedrin, as the Ran wrote — that he has authority to act outside the formal law in order to repair breaches. When monarchy ceased, that authority passed to the Sanhedrin; again, by simple reasoning, because when one governing institution is abolished, another institution takes its powers into its hands. I wrote about this in Column 101 and in more detail in Column 164. See there for halakhic implications of this historical accident and our collective lack of awareness of it (for example, the laws governing how communities are run in the Shulchan Arukh, which have nothing whatsoever to do with Jewish law in the strict sense).