This is generally not true; the overhead charges from grants and research money just about pay for the added cost of the (much) higher faculty salaries, travel, research equipment, and the increasingly onerous grant application and accounting/auditing/reporting processes, but not for undergraduate education. In fact it's usually prohibited to cross-subsidize education with research grants, and funding agencies are surprisingly strict about this, doing things like auditing whether computers bought on a grant are used in teaching a course (not allowed).

In many cases grant overhead doesn't even pay for the cost of research; only the top-tier research universities are particularly successful at pulling in enough, and big enough, grants to do that (see this from Georgia Tech's former Computer Science dean: http://innovate-wwc.com/2011/05/18/if-you-have-to-ask-ten-su...).

The liberal arts, meanwhile, are so cheap to teach that in many cases they are actually making money, i.e. their expenses are lower than tuition fees, and the remaining tuition is used to cross-subsidize science education: http://www.today.ucla.edu/portal/ut/bottom-line-shows-humani...