Metal-dielectric-graphene hybrid heterostructures based on oxides Al2O3, HfO2, and ZrO2 as well as on complementary metal–oxide–semiconductor compatible dielectric Si3N4 covering plasmonic metals Cu and Ag have been fabricated and studied. We show that the characteristics of these heterostructures are important for surface plasmon resonance biosensing (such as minimum reflectivity, sharp phase changes, resonance full width at half minimum and resonance sensitivity to refractive index unit (RIU) changes) can be significantly improved by adding dielectric/graphene layers. We demonstrate maximum plasmon resonance spectral sensitivity of more than 30,000 nm/RIU for Cu/Al2O3 (ZrO2, Si3N4), Ag/Si3N4 bilayers and Cu/dielectric/graphene three-layers for near-infrared wavelengths. The sensitivities of the fabricated heterostructures were ~ 5–8 times higher than those of bare Cu or Ag thin films. We also found that the width of the plasmon resonance reflectivity curves can be reduced by adding dielectric/graphene layers. An unexpected blueshift of the plasmon resonance spectral position was observed after covering noble metals with high-index dielectric/graphene heterostructures. We suggest that the observed blueshift and a large enhancement of surface plasmon resonance sensitivity in metal-dielectric-graphene hybrid heterostructures are produced by stationary surface dipoles which generate a strong electric field concentrated at the very thin top dielectric/graphene layer.