Based on works by H. Weinberger, R. Hamilton and L. Evans, we obtain a strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds. This maximum principle in the most general form, is sharp and generalizes the classical Hopf strong maximum principle for elliptic operators of second order. We use this maximum principle to give some applications in Geometric Analysis. In particular, we obtain various Bernstein type results for higher co-dimensional graphs generated from maps between Riemannian manifolds. This is a joint work with K. Smoczyk