Moving Target Defense (MTD) is commonly used in changing the asymmetric situation between attack and defense in cyberspace. Route mutation is a vital branch of MTD. Current route mutation methods still have some limitations on unpredictability of mutating routing paths and time overhead. In this paper, we propose a Dynamic Random Route Mutation (DRRM) mechanism in order to maximize the unpredictability of mutation and reduce the time overhead. First, we adopt Jaccard distance matrix and temporal constraint to improve mutation space. Mutation space can be dynamically shifted to enhance the unpredictability. Then, we weaken attackers' capabilities of launching effective eavesdropping at a certain node by using route weight. Finally, we construct a pre-distribution strategy to reduce the transmission latency caused by route mutation. Simulation results show that DRRM not only raises the difficulty of network eavesdropping, but also ensures low mutation time overhead.