In this work, we demonstrate, theoretically and experimentally, a hybrid dielectric-plasmonic multifunctional structure able to provide full control of the emission properties of CsPbI3 perovskite nanocrystals (PNCs). The device consists of a hyperbolic metamaterial (HMM) composed of alternating thin metal (Ag) and dielectric (LiF) layers, covered by TiO2 spherical MIE nanoresonators (i.e., the nanoantenna). An optimum HMM leads to a certain Purcell effect, i.e., an increase in the exciton radiative rate, but the emission intensity is reduced due to the presence of metal in the HMM. The incorporation of TiO2 nanoresonators deposited on the top of the HMM is able to counteract such an undesirable intensity reduction by the coupling between the exciton and the MIE modes of the dielectric nanoantenna. More importantly, MIE nanoresonators result in a preferential light emission towards the normal direction to the HMM plane, increasing the collected signal by more than one order of magnitude together with a further increase in the Purcell factor. These results will be useful in quantum information applications involving single emitters based on PNCs together with a high exciton emission rate and intensity.