Autonomous landing of micro unmanned aerial vehicles (UAVs) on moving targets has the potential to resolve many limitations of small-scale UAVs, such as uninterrupted flight tasks, rapid deployment and recovery of multiple UAVs, and extended operational ranges through mobile recharging stations. In this work, we present and experimentally verify a new vision-based method that enables a micro UAV to land autonomously on a mobile landing platform. Our method, which can be implemented on small-scale UAVs with limited payload capabilities and computational resources, incorporates model predictive control, vision-based localization, and extended Kalman filter for path following, navigation, and guidance. Our method uses a closed-loop controlled gimbaled camera for visual navigation and relative localization of the landing platform, a sensor fusion technique based on extended Kalman filters for target localization, and a model predictive control scheme for autonomous landing of the UAV under system uncertainties and wind disturbances. We demonstrate flight experiments of autonomous landing with an average error of 39 cm from the center of a mobile platform.