The multi-term time-fractional diffusion equation is a useful tool in the modeling of complex systems. This paper aims to develop a high order numerical method for solving multi-term time-fractional diffusion equations. Based on the space-time spectral method, a high-order scheme is proposed in the present paper. In this method, the Legendre polynomials are adopted in temporal discretization and the Fourier-like basis functions are constructed for the spatial discretization. Such a space-time spectral method possesses high efficiency and exponential decay in both time and space directions. Rigorous proofs are given here for the stability and convergence of the scheme. Numerical results show good agreement with the theoretical analysis.