This study develops an effective numerical method for addressing the time-fractional gas dynamics equation formulated with the Caputo time-fractional derivative. Novel basis functions are utilized, formulated as particular generalized Fibonacci polynomials contingent on a free parameter. This family generalizes the second kind of Chebyshev family. For the proposed polynomials, we establish basic analytical properties, including closed-form series expansion, inverse relation, moment and linearization formulas, and operational matrices for both integer-order and Caputo fractional derivatives. Using these tools, the fractional model, together with its underlying conditions, can be transformed into a finite system of nonlinear algebraic equations via a collocation strategy. Using Newton’s iterative method, the resulting system can be treated. A full convergence analysis of the double generalized Chebyshev expansion is provided. We demonstrate the accuracy and reliability of the presented method through several numerical simulations. Comparisons with existing numerical methods show that this approach achieves higher accuracy and faster execution.