The tempered fractional diffusion equation is a generalization of the standard fractional diffusion equation that includes the truncation effects inherent to finite-size physical domains. As such, that equation better describes anomalous transport processes occurring in realistic complex systems. To broaden the range of applicability of tempered fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we have developed a pseudospectral scheme to discretize the space-time fractional diffusion equation with exponential tempering in both space and time. The model solution is expanded in both space and time in terms of Chebyshev polynomials and the discrete equations are obtained with the Galerkin method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed Chebyshev pseudospectral method yields an exponential rate of convergence when the solution is smooth and allows a great flexibility to simultaneously handle fractional time and space derivatives with different levels of truncation. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.