We propose the novel numerical scheme for solution of the multidimensional Fokker-Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.