If that company can sponser H1B from your Bachelor's or Masters, I think you must be able to work. even if the job is not related to your field, and if you really want that job and Vice Presedent really wants you, they can change the title of the Position which will look like the the position from your field.
Also there is a possibiltiy that you might have studied some subjects related with that field as well,,hover over your transcript ..