What does "but" imply in this sentence?

From the very second paragraph of "Foundation" by Isaac Asimov:

There were nearly twenty-five million inhabited planets in the Galaxy then, and not one but owed allegiance to the Empire whose seat was on Trantor.

I presume he means that they all owed allegiance to the Empire, but that phrasing sounds like he's saying the opposite. I want to read it as if 'but' is the subject of the sentence meaning 'exception'.

Your presumption is correct. I don't often hear "not one but" used this way in spoken English, but it's not terribly uncommon in written English. (Unfortunately, Google Ngram Viewer is no help here because it offers no way to distinguish between not one but verbed [as in your quote] and not one but number [not one, but two hippos in my swimming pool].)

Anyway, you can think of not one but with a past tense verb as equivalent to not one that didn't with a present tense verb:

There were nearly twenty-five million inhabited planets in the Galaxy then, and not one that didn't owe allegiance to the Empire whose seat was on Trantor.

You're correct about its meaning.

IF "but" is not making sense in this sentence, try substituting some words in:

...inhabited planets in the Galaxy then, and not one planet rebelled, but all owed allegiance to the Empire ...

Hope that makes it clearer.