Yet another one where the definition you're referring to isn't the only use of the word. It makes perfect sense using a perfectly acceptable use of the word prove:
The Old English interpretation[edit]
The phrase is also sometimes, though with less reliability, claimed to mean "the exception tests the rule," since the original meaning of "to prove" in Old English (originating from the Latin word probat, compare probe) was closer to "to test." This original use of "to prove" meaning "to test" can also be seen in the phrase "proof spirit" which was liquor which had been tested and shown to be of appropriate quality. This usage still exists a site designated for testing equipment or weaponry may be called a "proving ground" (most notably in the US Military). A firearm cartridge, or round, designed to produce pressures higher than what a firearm is rated for is called a "proof round" Firearms are required to be able to fire a "proof round" without failure to ensure they are safe to use with standard pressure rounds. A further example, used both metaphorically and literally, is the term bulletproof.
In some cases, either of these definitions of "prove" create a working sentence, which somewhat explains the change in definition. For example I can say that porn found on a priest's hard drive "proves" that he is a paedophile, and it works for either definition; it either (a) provides hard evidence of paedophilia, thus proving the accusation to be true in the modern sense of the word, or (b) provides a legitimate test of whether or not the priest is a paedophile if the subjects of the porn are under age, he is defined as a paedophile. However, the aphorism "the exception that proves the rule" was not treated properly by the shift in definition, and now it looks like a way for idiots to justify their idiocy.
Hence the phrase can be used correctly in the context of the scientific method, when testing a hypothesis by examining possible exceptions to it and whether they invalidate the hypothesis. So if I hypothesise that "all swans are white" (under the condition that at least one swan exists), the discovery of the black swan (Cygnus atratus) "proves" (meaning "tests") my rule. In this case, the rule is falsified.