The effect of an external in-plane electric field on neutral and charged exciton states in two-dimensional (2D) materials is theoretically investigated. These states are argued to be strongly bound, so that electron-hole dissociation is not observed up to high electric field intensities. Trions in the anisotropic case of monolayer phosphorene are demonstrated to be especially robust under electric fields, so that fields as high as 100 kV/cm yield no significant effect on the trion binding energy or probability density distribution. Polarizabilities of excitons are obtained from the parabolicity of numerically calculated Stark shifts. For trions, a fourth order Stark shift is observed, which enables the experimental verification of hyperpolarizability in 2D materials, as observed in the highly excited states of the Rydberg series of atoms and ions.