Tuned liquid dampers (TLDs) have been broadly applied to suppress structural vibrations. In the present study, a novel vibration mitigation device consisting of non-Newtonian fluids coupled with an elastic baffle is proposed. The fluid-structure interaction is studied numerically. To optimize the system, different fluids, including the Bingham fluid, the Pseudoplastic fluid, and the Dilatant fluid are used as the damping fluids and the vibration suppression ability of each fluid is studied. Moreover, the energy dissipation mechanisms of different liquids are obtained. The results show that the optimal vibration suppression in the container without a baffle can be achieved by using the Bingham fluid. In this case, the average amplitude decay rate of the container is 12.662% with about 0.199% improvement in the damping ratio when compared to water. In the container with an elastic baffle, however, both the Pseudoplastic fluid and the Dilatant fluid outperform water in the damping capacity. The average amplitude decay rates of these fluids are 50.960% and 43.794%, respectively. Moreover, their damping ratios are 0.035% and 0.019% higher than that of water, respectively.