A numerical method is proposed to solve the full-Eulerian time-dependent Vlasov-Poisson system in high dimension. The algorithm relies on the construction of a tensor decomposition of the solution whose rank is adapted at each time step. This decomposition is obtained through the use of an efficient modified Progressive Generalized Decomposition (PGD) method, whose convergence is proved. We suggest in addition a symplectic time-discretization splitting scheme that preserves the Hamiltonian properties of the system. This scheme is naturally obtained by considering the tensor structure of the approximation. The efficiency of our approach is illustrated through time-dependent 2D-2D numerical examples.