Global stability of autonomous polynomial differential systems is an important issue in applications. By means of the Lyapunov direct method, an improved algebraic approach to proving the global stability of such systems in the positive cone is presented in this paper. This approach is based on an important finding that the derivative of a Volterra-type Lyapunov function along a polynomial differential system can be expressed as a linear combination of some Volterra-type functions. Applications to some epidemic models demonstrate the effectiveness and universality of the approach. Moreover, according to the property of the Volterra-type function and the relation between the endemic equilibrium and the disease-free equilibrium, the applicable Lyapunov function for the disease-free equilibrium can be derived by reformulating the Lyapunov function for the endemic equilibrium. The approach proposed here greatly simplifies the discussion on global stability of the feasible equilibria of some models described by polynomial differential systems.