Accessing hadronic form factors at large momentum transfers has traditionally presented a challenge for lattice QCD simulations. Here, we demonstrate how a novel implementation of the Feynman-Hellmann method can be employed to calculate hadronic form factors in lattice QCD at momenta much higher than previously accessible. Our simulations are performed on a single set of gauge configurations with three flavors of degenerate mass quarks corresponding to mπ≈470  MeV. We are able to determine the electromagnetic form factors of the pion and nucleon up to approximately 6 GeV2, with results for the ratio of the electric and magnetic form factors of the proton at our simulated quark mass agreeing well with experimental results.