Computer models are widely used to simulate real processes. Within the computer model, there always exist some parameters which are unobservable in the real process but need to be specified in the model. The procedure to adjust these unknown parameters in order to fit the model to observed data and improve predictive capability is known as calibration. Practically, calibration is typically done manually. In this paper, we propose an effective and efficient algorithm based on the stochastic approximation (SA) approach that can be easily automated. We first demonstrate the feasibility of applying stochastic approximation to stochastic computer model calibration and apply it to three stochastic simulation models. We compare our proposed SA approach with another direct calibration search method, the genetic algorithm. The results indicate that our proposed SA approach performs equally as well in terms of accuracy and significantly better in terms of computational search time. We further consider the calibration parameter uncertainty in the subsequent application of the calibrated model and propose an approach to quantify it using asymptotic approximations.