In this study, a novel two-dimensional finite-volume method on unstructured triangular meshes for two-layer shallow water flows is developed. First, the one-dimensional relaxation approach developed in Abgrall and Karni (2009) [1] is extended to the currently studied two-dimensional two-layer shallow water equations to build an unconditionally hyperbolic system. Next, a two-dimensional finite-volume HLL scheme on triangular grids for such a non-conservative hyperbolic system is constructed. Special attention is paid to guaranteeing the well-balanced property even in the presence of the wet-dry fronts. To this end, techniques such as spatial reconstruction for the secondary variables, special cell classifications for the two-layer fluid system, and special local reconstruction of auxiliary surface variables are proposed. These special techniques are further combined with new discrete formulas for the nonconservative products and the bed-slope terms, and it thus leads to an exactly well-balanced numerical scheme for the studied two-layer system even in the presence of wet-dry fronts. The developed numerical model is second order accurate and preserves the nonnegative layer depths. The performances of the proposed numerical model are investigated by several numerical experiments.