I've found something else that may be even better: WebCite. It allows you to trigger archiving of a web page at the time you cite it.
Note that WebCite "honors robot exclusion standards, as well as no-cache and no-archive tags", so if someone doesn't want their page archived for any reason, it will be excluded. It looks like they'll also remove a page from their archive upon request. (See their FAQ.) But I think most dead links are just because someone stopped maintaining their page.