Slow down. Read the question carefully, and identify any elements that you understand or recognize. You've been in class all semester, you've done the reading, and you studied for this exam. You are sure to know something about the topic even if you don't know how to answer the question.
Don't commit yourself to anything you don't know. Essay questions are great because you get to discuss broader concepts. They aren't true or false, so you have some wiggle room. Use it to your advantage. Never define something, cite a date or name a place unless you are absolutely positive you are spot on. One completely off point or wrong fact, and you totally blow your cover.
Add something concrete. Draw on what you do know to legitimize your essay. Even if you don't think you can directly address the question, pulling in examples from your assigned reading, course projects or class discussion can go a long way towards making you sound like you know what you're talking about.
Provide an easy framework for yourself and your argument. Fall back on the classic five-paragraph structure with a clear introduction, body and conclusion. Use a clear structure to help you avoid running off on tangents and losing focus, both of which are clear signs that you don't have a solid to answer the question. A defined structure can also help you gauge your time more accurately.
Argue logically. Even if you are slightly off your teacher's topic, if your essay flows logically and stays focused on the point you're arguing, then you'll sound somewhat credible. Definitely avoid fallacies and angry, personal diatribes.
Stroke your teacher's ego. Most teachers have pet terms and phrases that they frequently use to discuss their favorite topics. If you can, subtly parrot back your teacher's words in your essay, changing them just slightly so that they don't sound like quotations. If, however, you can't think of anything relevant to use, don't try it. Your teacher may end up seeing right through you.
Make your essay perfect. You can earn yourself easy points if your paper is grammatically perfect with no spelling or punctuation errors, even if you didn't exactly answer the question. If the essay instructions list any other requirements, make sure you follow them to the letter. In your situation, you really can't afford to make simple mistakes. Don't give your teacher any other excuse to mark your paper down.