Herein, we propose new efficient spectral algorithms for handling the fractional diffusion wave equation (FDWE) and fractional diffusion wave equation with damping (FDWED). In these algorithms, we employ new basis functions of the shifted fifth-kind Chebyshev polynomials that satisfy all the initial and boundary conditions of the equation. The key idea of the presented algorithms depends on transforming the FDWE and FDWED with their underlying conditions into systems of algebraic equations in the unknown expansion coefficients. Our study is supported by a careful convergence analysis of the suggested shifted fifth-kind Chebyshev expansion. Finally, some numerical examples are presented to confirm the accuracy and efficiency of the proposed algorithms.