This was a very short short story, a few pages IIRC. I read it recently (within the past five years) in English, although of course the story could have been older and it could have been a translation. I think I read it in an anthology that I got from the library, but I have gone through my check-out history with my library and had no luck.
Likewise no luck with Google, DuckDuckGo, and TVTropes.
I think there was a foreshadowing of the reveal when the leader of the crew claims the planet by burying a bone on its surface, and the last line of the story is something like "He [the leader of the crew] knew he was a Good Dog".
Edited to add: I don't recall what happened to eradicate the human race in the story, but I do remember that the dogs talk (right before that last line) about claiming the planets to have habitable worlds ready for the humans' return. That's why the leader is a "Good Dog" - he's doing his duty for humankind.