The existence of a "knee" at energy ∼1 PeV in the cosmic-ray spectrum suggests the presence of Galactic PeV proton accelerators called "PeVatrons." Supernova remnant (SNR) G106.3+2.7 is a prime candidate for one of these. The recent detection of very high energy (0.1-100 TeV) gamma rays from G106.3+2.7 may be explained either by the decay of neutral pions or inverse Compton scattering by relativistic electrons. We report an analysis of 12 years of Fermi-LAT gamma-ray data that shows that the GeV-TeV gamma-ray spectrum is much harder and requires a different total electron energy than the radio and x-ray spectra, suggesting it has a distinct, hadronic origin. The nondetection of gamma rays below 10 GeV implies additional constraints on the relativistic electron spectrum. A hadronic interpretation of the observed gamma rays is strongly supported. This observation confirms the long-sought connection between Galactic PeVatrons and SNRs. Moreover, it suggests that G106.3+2.7 could be the brightest member of a new population of SNRs whose gamma-ray energy flux peaks at TeV energies. Such a population may contribute to the cosmic-ray knee and be revealed by future very high energy gamma-ray detectors.