Thankfully the Black Widow is one of the dumbest spiders I've every encountered. Amazingly easy to kill if you can be calm about it. (I feel like I should be knocking on wood now.) I don't know how many varieties there are though, so YMMV.
I had no idea the bite was that bad or I would have just finished it off instead.
Pretty much as soon as I find one I immediately drop whatever I'm doing to make sure it's out of my living space as quickly as possible before I lose track of it.