We introduce an automatic variationally stable (AVS) finite element (FE) method for the numerical analysis of convection-dominated second order boundary value problems. The method is a hybrid continuous-discontinuous Petrov-Galerkin (DPG) method and uses standard FE trial functions that are piecewise globally continuous polynomials. The test functions, however, are piecewise discontinuous and computed by using the DPG approach resulting in unconditionally stable FE discretization. Remarkably, the support of each discontinuous test function is identical to its corresponding continuous trial function and the element restrictions of the test functions contribution can be computed completely locally (i.e. decoupled). Additionally, numerical evidence indicates that the computation of the (optimal) test functions is achieved with sufficient accuracy by using the same polynomial order of approximation as used for the trial functions. As in every other DPG formulation, the resulting algebraic system is symmetric and positive definite, allowing to use simple iterative strategies to compute the numerical solution. Goal oriented estimates of the numerical approximation error can subsequently be obtained by applying a residual-based error estimation technique to the AVS-FE method. Numerical examples are shown for convection-dominated diffusion problems.