A moment method known as the fourth moment method can perform reliability analysis without optimization using the first four statistical moments. Numerical integration is used to calculate the statistical moments, where a moment-based quadrature rule can be used for integration nodes and weights. However, the moment-based quadrature rule has to solve a system of linear equations that can be numerically unstable. Considering this point, an improved moment-based quadrature rule is proposed and is applied to reliability-based design optimization. Finally, the moment-based RBDO is applied to numerical examples with a variety of random variables and target reliability indexes. From the numerical results, the performance of the improved moment-based quadrature rule can be confirmed and several guidelines are given for the moment-based RBDO.