In this paper we demonstrate an application of nonsmooth analysis and the theory of exact penalty functions to two-dimensional problems of the calculus of variations. We derive necessary conditions for an extremum in the problem under consideration and use them to construct a new direct numerical minimization method (the method of steepest descent). We prove the convergence of the method and give numerical examples that show the efficiency of the suggested method.The described method can be very useful for solving various practical problems of mechanics, mathematical physics and calculus of variations.