A divide-and-conquer solver coupled with Tensor-Train (TT) format is proposed for solving the d-dimensional time-space fractional diffusion equations with alternating direction implicit finite difference scheme. In contrast to the complexity and storage of divide-and-conquer solver in full storage format, which grows at least exponentially with dimension d, complexity and storage of the proposed solver are O(NM(log M+ r) (r+ (d- 1) r) + Nlog N· r) and O(M(r+ (d- 1) r) + Nr) with acceleration by an efficient approximated Toeplitz inversion, where M, N are numbers of spatial and temporal grid points and r is the TT-ranks of related TT format data. Hence they grow slowly with dimension d if the TT-ranks r is low. For reducing the increase of TT-ranks after certain TT operations, TT rounding is performed with introduced error, which is analyzed for giving a criterion for preserving convergence rate of the numerical scheme. The accuracy and efficiency of the proposed solver is illustrated by numerical experiments with dimension d up to 20 for case of low TT-ranks initial conditions and source terms.