In this paper, we propose an efficient and accurate numerical method for computing the dynamics of rotating two-component Bose-Einstein condensates (BECs) which is described by the coupled Gross-Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field. By introducing rotating Lagrangian coordinates, we eliminate the angular momentum rotation term from the CGPEs, which allows us to develop an efficient numerical method. Our method has spectral accuracy in all spatial dimensions and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in the literature. Numerical results show that to achieve the same accuracy, our method takes much shorter computing time. We also apply our method to study issues such as dynamics of vortex lattices and giant vortices in rotating two-component BECs. Furthermore, we generalize our method to solve the vector Gross-Pitaevskii equations (VGPEs) which is used to study rotating multi-component BECs.