Is eliminating foods I'm allergic to from my diet enough to prevent an averse reaction?

In most cases, yes. An elimination diet is an effective way of avoiding allergic reactions to foods. However, your physician should ascertain that you are allergic. A positive skin or blood test for allergy is not enough. You also need to have a suffered a reaction that correlates with that test (for instance developing hives when you eat peanuts).
Yes. If you are allergic to a specific food (i.e. Ige antibodies to a protein), avoidance is crucial to remain free of reactions. However, always carry your Epipen (epinephrine) with you as well as some antihistamine for accidental exposures.