This issue encompasses a "British/American" difference, and apparently, an evolving set of definitions.
Undergraduate degree has come to mean, at least in the US, the degree one obtains upon completing an undergraduate program (see below for graduate/postgraduate UK/N. America disambiguation).
I assume that this is a newer usage, at logical odds with the standard definition of undergraduate as, well, being under graduation: not yet having graduated. But it yields to logic from another perspective: one gets a graduate degree from a graduate program/graduate school, and an undergraduate degree from an undergrad program.
I was program coordinator, undergraduate studies at a major US university, and can attest that this is the terminology most commonly used among higher education administrators and related professionals and organizations (federal and state education officials, etc.) in the US for "first degrees" in higher education.
Wikipedia entry for "undergraduate degree":
An undergraduate degree (also called first degree, bachelor's degree, or simply degree) is a colloquial term for....
Wikipedia entry for "postgraduate education":
Postgraduate education (or graduate education in North America)....
Dictionaries have not caught up with this (US) usage, but it is quite established.