Using multiband k·p calculations, we show that strain-engineered piezoelectricity is a powerful tool to modulate the electron-hole spatial separation in a wide class of wurtzite CdSe/CdS nanocrystals. The inherent anisotropy of the hexagonal crystal structure leads to anisotropic strain and, consequently, to a pronounced piezoelectric field along the c axis, which can be amplified or quenched through a proper design of the core-shell structure. The use of large cores and thick shells promotes a gradual departure from quantum confined nanocrystals to a regime dominated by piezoelectric confinement. This allows excitons to evolve from the usual type-I and quasi-type-II behavior to a type-II behavior in dot-in-dots, dot-in-rods, rod-in-rods, and dot-in-plates. Piezoelectric fields explain experimental observations for giant-shell nanocrystals, whose time-resolved photoluminescence reveals long exciton lifetimes for large cores, contrary to the expectations of standard quantum confinement models. They also explain the large differences in exciton lifetimes reported for different classes of CdSe/CdS nanocrystals.