I read this in a 70's or 80's mass market paperback anthology. I believe it may have been a single-author anthology, but I'm not sure.
The cover may have had a statue on it of the extinct god-figure "Man".
The story concerns the last man on Earth, who knows that he will soon die. At some point during Man's long decline, he raised dogs to sentience. Dogs are almost extinct too, though - there is one male dog and a handful of female dogs left. The dogs have been sterilized, to control them, and the man controls the antidote.
The male dog, who has been the man's friend, asks him for the antidote so that the dogs can keep going after humans are finally extinct. The man considers this, and imagines the future that the dogs will build after he is gone. He concludes that because of the positive qualities dogs possess, sentient dogs will eventually build a society better than anything humanity ever built.
He then lies to the dog and tells him he will give him the antidote - but throws it over a cliff. Since dogs - even sentient dogs - can't resist playing fetch, the male dog jumps over the cliff after it. The female dogs wail and howl because they know the man has maliciously stolen the future from them, because he couldn't stand the thought of dogs being better than humanity.
I have found this to be a difficult story to search for, because no matter how I phrase the story elements I end up getting a hundred hits for Simak's City, which is not it.