We perform a nonperturbative lattice calculation of the complex phase and modulus of the pion form factor in the timelike momentum region using the finite-volume technique. We use two ensembles of 2 + 1 - flavor overlap fermions at pion masses m(pi) = 380 and 290 MeV. By calculating the I = 1 correlators in the center-of-mass and three moving frames, we obtain the form factor at ten different values of the timelike momentum transfer around the vector resonance. We compare the results with the phenomenological model of Gounaris-Sakurai and its variant.