In the footage, Chris Wallace questioned Blunt on legislation that benefited the Philip Morris tobacco company. Fox said it was made to appear like an endorsement of Carnahan.
Blunt won the election.
In a joint statement issued Friday, the campaign acknowledged that the clip "exceeded" the bounds of the fair-use doctrine but did not pay Fox any money, The Kansas City (Mo.) Star reported.
Fox lawyer Bernie Rhodes said the network succeeded in "protecting our rights and keeping people from doing this."
A Carnahan spokesman claimed victory because the suit was ended without money changing hands.