Term for completion of a Post-graduate degree

Yes, it is correct and common, at least in the United States, to say "he/she has graduated" when referring to a Masters or PhD.

To distinguish it from a Bachelors degree, it is common to hear

"He/she graduated with a Master's in Mathematics."

or

"He/she has graduated with a PhD."

The term "graduate" is largely used when referring to students who have graduated with a bachelors, masters, or doctoral degree. There really is no specific term (equivalent of graduate) that specifies the degree a student received. If you only need to convey that someone graduated from college, you can just say "she is a college graduate). If you want to provide information regarding the level of the degree earned, you would just have to specify in the sentence (e.g. She earned a Masters degree in Agriculture from Louisiana State University). You wouldn't need to state that she was a graduate with a masters degree, because it is already implied.

Now, if you want to refer to someone who already received a bachelors degree and is still in the process of receiving a higher degree, the appropriate term is "graduate student." This does not have the same meaning as the term "graduate." For example, I am currently working on my Phd. Therefore, you would refer to me as a graduate student.

Hope that makes sense/answers your question to some degree [no pun intended ;)]