High-harmonic generation in condensed-matter systems is both a source of fundamental insight into quantum electron motion and a promising candidate to realize compact ultraviolet and ultrafast light sources. While graphene is anticipated to efficiently generate high-order harmonics due to its anharmonic charge-carrier dispersion, experiments performed on extended samples using THz illumination have revealed only a weak effect. The situation is further complicated by the enormous electromagnetic field intensities required by this highly nonperturbative nonlinear optical phenomenon. Here we argue that the large light intensity required for high-harmonic generation to occur can be reached by exploiting localized plasmons in doped graphene nanostructures. We demonstrate through rigorous time-domain simulations that the synergistic combination of strong plasmonic near-field enhancement and a pronounced intrinsic nonlinearity result in efficient broadband high-harmonic generation within a single material. Our results support the strong potential of nanostructured graphene as a robust, electrically tunable platform for high-harmonic generation.