In general, dielectric multifiller polymer composites have the potential to achieve enhanced performances by integrating the desirable properties of each filler. However, the improvement in thermophysical and dielectric properties is often accompanied by a deterioration of electrical breakdown strength (EBD). Here, we explore a two-filler polymer nanocomposite structure, based on polyolefins with montmorillonite and calcium carbonate fillers, and present an effective approach to obtain enhanced EBD by tailoring the composite morphology (by designing the pseudo-two-dimensional nanoclays to preferentially physisorb on the surfaces of calcium carbonates, so as to change the nature of the filler/polymer interfaces). It is shown that, in these structured polymer nanocomposites, the breakdown performance is substantially improved, exceeding the performance of the unfilled polymers and of the respective single-filler composites. The enhanced dielectric behavior originates from the specific composite morphology, which capitalizes on the platelet nanofillers to enhance the microfiller/polymer interfaces and on the extended hybrid-filler structure to mitigate low-energy failure initiation and propagation.