Since rats were one of two vermin (the other being cockroaches) proven tough and hardy enough to survive and recolonize Pacific isands and atolls subjected to the blast, heat, and radiation of nuclear weapons tests, the most we can aspire to is a holding action - unless we get as deadly serious as some parts of Canada have been (although local climate admittedly plays a part in their success thus far). 

I think if humanity ends up exterminating itself, lamentably, it'll be a distant descendant of a rat or cockroach that will be left to reinvent civilization.