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 m(pi) approximate to 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.