Control and autonomy of microswarms have drawn increasing attention in recent years. Especially in dynamic environments, robust navigation of swarms avoiding obstacles in real-time still remains challenging. To tackle this issue, in this work we developed an automatic navigation scheme for microswarms, including a fast path planning module and a robust motion control module. At first, we designed a discrete RRT* (d-RRT*) algorithm with enhanced searching efficiency and space availability to guarantee the real-time requirement. Then, we presented a disturbance observer (DOB) based super-twisting sliding mode controller (STSMC) to govern the trajectory following task against external disturbances and system uncertainties. Finally, we performed simulations and experiments to validate the proposed scheme. Results indicate that the d-RRT* algorithm could uniformly explore the working environments and provide a faster planning speed compared to conventional RRT* planner, and the DOB-STSMC could guarantee a tracking error within half swarm body length. Furthermore, our method successfully navigated a vortex-like microswarm to the target position while avoiding two dynamic virtual obstacles.