An algorithm for the numerical solution of low-thrust Lambert's problem is proposed. After averaging the extremal flow of the optimal control Hamiltonian, a one-parameter family of solutions of a reduced-order two-point boundary value problems is achieved by means of a differential continuation scheme. Sensitivities of the shooting function are then used in conjunction with an ad hoc near-identity transformation between averaged and osculating variables to achieve an accurate solution for all longitudes of the departure and arrival orbits. Hence, a single simplified shooting problem has to be solved to approximate the solution for any combination of departure and arrival dates (i.e., to draw a so-called "pork-chop chart"). Both the averaged flow and the near identity transformation are efficiently evaluated via the fast Fourier transform algorithm, yielding a fully numerical procedure.