The sentence from the NYT is fine. Here's why.
Living beneath the ground, scientists discovered a bat colony.
In general, when
a writer an English-speaker starts a sentence with a participial phrase and proceeds immediately to the subject, he means that the subject is doing that participle. (There are exceptions, but the participles are unattached to the other words in the sentence.) Here, the scientists are living underground. The sentence above is grammatical; we know, though, that that's not what the writer meant. In other words, the sentence is unambiguous; it just happens to mean something other than what was intended.
On the other hand, I'll allow myself to say that a sentence like
We saw a calico cat driving down the road.
is ambiguous. Again, we're pretty sure what the writer means. However, there are plenty of sentences where the participle heading the final phrase is supposed to be attached to the noun right before it; maybe the cat was driving? That said, I'm not sure I'd view something like
We saw a majestic redwood driving down the road.
as equally ambiguous. But the fact that ambiguity might appear when a participial phrase finishes a sentence but can't appear when it starts it suggests that there's something beyond put-it-next-to-what-it-modifies going on.
The phrase living beneath the ground is adjectival and it modifies bat colony. In the cat example, though, driving down the road might be adjectival, in which case it modifies cat, or adverbial, in which case it modifies the verb. Note that I can also write
We saw, driving down the road, a calico cat hugging a majestic redwood.
I didn't realize it until after I wrote it, but I almost shot myself in the foot there with that hugging a majestic redwood. But that can't modify the verb. Or at least, it's really implausible. Often, the phrase will offer some sort of context for the verb, like a frame around it. It makes sense that driving can offer the context for us seeing something; it makes less sense that hugging could do the same.
Rather, that example is actually a nice segue to my main point.
Petro O. Poroshenko is president of Ukraine, promising to put an end to a separatist insurrection in the east that has divided the country for months.
Promising to put an end to a separatist insurrection in the east that has divided the country for months, Petro O. Poroshenko is president of Ukraine.
How do you like those? They don't sound so good, do they? That's because the participial phrase is adverbial; it can't add anything to his being president. If, however, we want to explain in what manner he was sworn in i.e., if we want a verb that could meaningfully be modified it works fine. The original sentence
Petro O. Poroshenko, a pro-European billionaire confectioner, was sworn in on Saturday as the fifth president of Ukraine, promising to put an end to a separatist insurrection in the east that has divided the country for months.
has nothing wrong with it. I noticed that, in your question, you never mentioned anything about ambiguity; that was something a few others who found fault with the sentence added. It's not ambiguous because there's no investiture or appointment or anything else that the phrase could plausibly modify save the verb. (If anyone wants to say there's Poroshenko, I won't bother debating the point. Whether it's Poroshenko or the way Poroshenko acted in being sworn in is academic, at least as far as meaning is concerned.)
To answer your question, though: Can one end a sentence with a participial phrase? Yes. I wish I could offer some more sources, but I recently moved and haven't found everything. Here's something from Purdue; note that they don't say you'll be punctuating an ungrammatical sentence. And I'm kind of surprised to say that Uncle Teng's page has some of the best examples and explanations I was able to find on Google.