We present a theory for terahertz (THz) wave emission in a cavity containing a graphene-based asymmetric hyperbolic metamaterial (AHMM), and the results of the calculation of the output power of the THz laser. An asymmetry appears in the spatial spectrum of the eigenmodes due to a tilt of an anisotropy axis with respect to the interfaces of the AHMM that results in a difference in the wave vector components for waves propagating in opposite directions. This AHMM is an active medium in the THz region due to an inverted population of charge carriers in graphene. The theory is based on the analysis of eigenwaves in the cavity containing the graphene-based AHMM and the use of the transfer matrix method. The components of the Poynting vector, normal to the interface of the AHMM, have been calculated for different eigenmodes, taking into account the gain saturation. The gain saturation manifests itself as a decrease of the negative imaginary part of the effective permittivity of the metamaterial. This gain saturation results from the dependence of the chemical potential in graphene on the component of the electric field transverse to the graphene sheets. Here, the balance of gain versus loss predicts the intensity of the AHMM THz laser emission.