We present a new hybrid pressure formulation applied to the control volume finite element (CVFE) method to model multiphase flow and transport in highly heterogeneous porous media. The formulation effectively captures sharp saturation changes in the presence of discontinuous material properties by employing a discontinuous pressure approximation at material interfaces. The heterogeneous porous medium is divided into sub-domains within which material properties are uniform or smoothly varying. By construction, the resultant control volume dual mesh is restricted within a sub-domain. The artificial mass leakage across material property boundaries observed in classical CVFE methods is therefore circumvented. The approach applies the robust continuous pressure approximation in the rest of the computational domain; the discontinuous approximation is applied only at the sub-domain boundaries. The discontinuous parameters necessary to achieve mass conservative solutions, locally and globally, are described. We demonstrate the accuracy and efficiency of the new approach by comparison with the classical continuous CVFE method on various examples of heterogeneous domains as well as establishing the convergence of the numerical method. The proposed hybrid formulation significantly outperforms the accuracy and efficiency of classical CVFE methods that use the same order of approximation for modeling multiphase flow in heterogeneous porous media.