The first direct measurement of electron neutrino quasielastic and quasielasticlike scattering on hydrocarbon in the few-GeV region of incident neutrino energy has been carried out using the MINERvA detector in the NuMI beam at Fermilab. The flux-integrated differential cross sections in the electron production angle, electron energy, and Q^{2} are presented. The ratio of the quasielastic, flux-integrated differential cross section in Q^{2} for ν_{e} with that of similarly selected ν_{μ}-induced events from the same exposure is used to probe assumptions that underpin conventional treatments of charged-current ν_{e} interactions used by long-baseline neutrino oscillation experiments. The data are found to be consistent with lepton universality and are well described by the predictions of the neutrino event generator GENIE.