We report on the observation of two neutrino-induced events which have an estimated deposited energy in the IceCube detector of 1.04±0.16 and 1.14±0.17  PeV, respectively, the highest neutrino energies observed so far. These events are consistent with fully contained particle showers induced by neutral-current νe,μ,τ (ν¯e,μ,τ) or charged-current νe (ν¯e) interactions within the IceCube detector. The events were discovered in a search for ultrahigh energy neutrinos using data corresponding to 615.9 days effective live time. The expected number of atmospheric background is 0.082±0.004(stat)+0.041−0.057(syst). The probability of observing two or more candidate events under the atmospheric background-only hypothesis is 2.9×10−3 (2.8σ) taking into account the uncertainty on the expected number of background events. These two events could be a first indication of an astrophysical neutrino flux; the moderate significance, however, does not permit a definitive conclusion at this time.