According to a recent Associated Press article, "Some Republicans had argued that at the time of the election, Carnahan was not -- as the U.S. Constitution requires -- a Missouri inhabitant, because he was dead."
The precedents this sort of thing could set are mind-boggling. Could I campaign for the Senate on under the name of of, say, my late uncle Bob? What if a living Senator decided he didn't want to do the job any longer -- could he send his wife in finish his term up?
I suppose it's fair to say that the voters knew they were really voting for Carhanan's widow instead of for the man himself, but it wasn't her name on the ballot. I really don't see how she can be legally construed as the winner.
The question, though, is who else should replace Mr. Carnahan. I don't see how it can automatically be the Republican candidate, unless you consider him as having run unopposed. The other thing to do would be to interpret the 17th Amendment as applying to this case: "When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies", leaving it up to the new governor to choose the Senator's replacement.