This seems to be slightly confusing to me, because the regulations state that the job that a foreign student graduated from a Germany university can take "should be related to his/her field of study" which seems a bit vague. If I hold a Bachelor's degree in Computer Science (classification A3), but later do a master's degree in another field (Master of Arts), can I still land a job as software engineer/"IT specialist" using my bachelor's degree as qualification? I think there shouldn't be any reason that the job has to be restricted by the graduate degree only, but I'm not totally sure about it. Also, a personnel from DAAD in my region gave me some confusing replies as well(he told me to ask the diploma-offering university in Germany for clarification).
You write "the regulations state that the job that a foreign student graduated from a Germany university can take "should be related to his/her field of study". What regulations are that?
If the regulation under which you are currently allowed to live and study in Germany is § 16 Aufenthaltsgesetz (AufenthG) in your case, I cannot find any regulation as to what field you are allowed to work in. It just states that you may not work for more then 120 days / year as a general rule.
Believe it or not, law is more logical in Germany than one may think. Just get used to asking youself what the abuse of a law mihght be and you usually understand better what the idea behind a law is.
In this case it's: If they would allow you to work full-time 365 days a year, please could easily circumvent any working visa and just register as a student in a German university (which would by the way provide some more goodies) but never actually study at all. It would take at least one year, in practice rather 2-3 years before the university would kick you out and as a result you'd loose your visa. And this is what they don't want. On the other hand they don't want to limit to opportunity to study to those who bring enough money to sustain their living in Germany for several years without any income. Thus the compromise that they allow you to work for 120 days / years which will hopefully be enough to cover your cost and leaves enough time for actually studying.