This article proposes a novel Lyapunov stability analysis for a class of switched nonlinear systems under the mode-dependent average dwell time (MDADT) switching signals. For the first time, a polynomial fuzzy model is used to describe the nonlinearity of the switched systems instead of a Takagi–Sugeno fuzzy model, which can represent a wider range of nonlinear plants. As existing advanced results are relatively conservative and lead to that tighter bounds on the dwell time are not easy to be obtained, therefore, the stability analysis for switched systems is still quite challenging. In this article, to tackle this problem, by developing a polynomial multiple Lyapunov function (MLF) approach and exploring the feature of the membership functions, relaxed stability conditions are derived for the switched polynomial fuzzy-model-based control systems. In particular, the polynomial MLF is guaranteed radially unbounded by a two-step procedure, making traditional quadratic MLF a specific case, thereby improved stability conditions can be established in the form of sum-of-squares. Moreover, the new algorithm for multivariate Chebyshev membership functions based on linear programming is proposed to introduce the information of membership functions and further relax the stability conditions to obtain tighter bounds on MDADT. Finally, a simulation example demonstrates the effectiveness of the developed techniques.