I understand how you feel because I used to be the same way. Here are some suggestions that I found helpful for myself:
1. Stop telling myself that I am of the type to get nervous on tests. It plays to the idea of not defeating yourself before you start. Try to go into a test confident of your preparation/knowledge of the material. Looking at a test as merely a tool to diagnose how much you know in a subject is a better way to picture tests than as some indication of self worth. Which leads into my second tip..
2. Prepare properly for your exam. You said that you do well in class so this is already a plus and is a good indication that you are learning the material. After the first test or two, you should be able to pick up on your professor's style and anticipate the type of exams they give. Do they ask critical thinking questions? Are the problems similar to the homework or examples done in class? Make up practice exams to get you comfortable with test taking. This will also be an excellent way to prepare for exams.
3. Get a good night's sleep and decent breakfast/lunch ( depending on when your exam is

) Also, allow plenty of time to get to class and pick out a nice seat. Although this may sound funny, I often got to class at least 10 minutes early so that I could pick a favorable seat ( sometimes the favorability is completely psychological) and get used to my surroundings. Nothing is worse than rushing to class ( worried that you are going to be late for an exam) and getting there a minute before the exams are passed out and realize that you do not have a pencil, your calculator is dead, the only free seat is in the middle of a sea of people
4. During the exam, answer all the easy problems first and come back to the harder ones. I used to have the bad habit of getting stuck on a single problem and wasting away my time when I could have moved on and picked up several easy points. If you waste your time and then find yourself rushing at the end of the test, you increase your chances of getting simples problems that you would normally get correct wrong.
5. If you ever finish an exam early, use every minute to your disposal. I cannot even count the number of times that I found a simple error that was made absentmindedly that I caught on the second or third time through a test. Although you may find that a rare amount of exams will permit enough time to thouroughly double check your work, when you get the oppurtunity make sure to take it.
Hopefully these suggestions will help for you as they did for me. And remember, things won't magically change overnight. Stay consistent and results will show.