The article analyzes an upwind-mixed hybrid finite element approximation scheme for a linear parabolic advection-diffusion-reaction problem. The main goal of the work is to improve the robustness of the standard mixed method for strongly advection-dominated cases. Existence and uniqueness of a weak solution is proven first. The mixed hybrid formulation associated to the problem is introduced next and a fully discrete upwind-mixed hybrid scheme is given. The scheme the authors introduce is based on the Raviart-Thomas mixed finite element of the lowest order. The definition of the upwind weights involves inter-element multipliers which enforces continuity of the normal fluxes across element boundaries. These multipliers are introduced into the system by hybridization and the number of unknowns in the linear system is reduced due to the elimination of local variables. In the part of the article where the analysis of the scheme is detailed, the authors also give uniqueness and existence results for the scheme. Error analysis for the fully discrete upwind-mixed scheme is carried out and optimal order convergence in time and space is obtained. Numerical results given in the article confirm the sharpness of the error bounds derived by the analysis and also demonstrate the efficiency of the method.