The supersymmetric approach in the form of second-order intertwining relations is used to prove the exact solvability of the two-dimensional Schrödinger equation with generalized two-dimensional Morse potential for a0 =-1 2. This two-parametric model is not amenable to the conventional separation of variables, but it is completely integrable: the symmetry operator of fourth order in momenta exists. All bound-state energies are found explicitly, and all corresponding wave functions are built analytically. By means of the shape invariance property, the result is extended to the hierarchy of Morse models with arbitrary integer and half-integer values ak =- (k+1) 2.