We consider a damped oscillator mode that is resonantly driven and is coupled to an arbitrary target system via the position quadrature operator. For such a composite open quantum system, we develop a numerical method to compute the reduced density matrix of the target system and the low-order moments of the quadrature operators. In this method, we solve the evolution equations for quantities related to moments of the quadrature operators, rather than for the density matrix elements as in the conventional approach. The application to an optomechanical setting shows that the new method can compute the correlation functions accurately with a significant reduction in the computational cost. Since the method does not involve any approximation in its abstract formulation itself, we investigate the numerical accuracy closely. This study reveals the numerical sensitivity of the new approach in certain parameter regimes. We find that this issue can be alleviated by using the position basis instead of the commonly used Fock basis.