Is there a university that has only graduate programs?
There are some universities that offer primarily postgraduate degrees; Wikipedia has a list. Some notable examples on this list are Rockefeller University (US) and the Weizmann Institute of Science (Israel).

However, this doesn't necessarily save money; undergraduates in the United States typically pay tuition, after all. Other sources of income for universities in the United States (such as federal Pell grants) also apply only to undergraduate students. Furthermore, having a local population of undergraduate students allows budding academics (PhD students) to get teaching experience.