Punctuation within quotes or outside?

Hello

In this sentence:

"It was a ghastly day," Smith said.

Does the comma after "day" belong outside the quote marks or inside? Inside seems right, intuitively, though we're told that punctuation that does not belong in the quoted material must be placed outside quotes. Here, the comma after "day" is not really part of the quote. Have I got it wrong? Or is there an exception to the rule that I've missed?

Thank you.