Fluctuations are inherent to any fabrication process. Integrated circuits and microelectromechanical systems are particularly affected by these variations, and due to high-quality requirements the effect on the devices' performance has to be understood quantitatively. In recent years, it has become possible to model the performance of such complex systems on the basis of design specifications, and model-based sensitivity analysis has made its way into industrial engineering. We show how an efficient Bayesian approach, using a Gaussian process prior, can replace the commonly used brute-force Monte Carlo scheme, making it possible to apply the analysis to computationally costly models. We introduce a number of global, statistically justified sensitivity measures for design analysis and optimization. Two models of integrated systems serve us as case studies to introduce the analysis and to assess its convergence properties. We show that the Bayesian Monte Carlo scheme can save costly simulation runs and can ensure a reliable accuracy of the analysis