In this paper, we investigate the use of dynamic structural tailoring via the concept of an Acoustic Black Hole (ABH) to enhance the performance of piezoelectric based energy harvesting from operational mechanical vibrations. The ABH is a variable thickness structural feature that can be embedded in the host structure allowing a smooth reduction of the phase velocity while minimizing the amplitude of reflected waves. The ABH thickness variation is typically designed according to power-law profiles. As a propagating wave enters the ABH, it is progressively slowed down while its wavelength is compressed. This effect results in structural areas with high energy density that can be exploited effectively for energy harvesting. The potential of ABH for energy harvesting is shown via a numerical study based on fully coupled finite element electromechanical models of an ABH tapered plate with surface mounted piezo-transducers. The performances of the novel design are evaluated by direct comparison with a non-tapered structure in terms of energy ratios and attenuation indices. Results show that the tailored structural design allows a drastic increase in the harvested energy both for steady state and transient excitation. Performance dependencies of key design parameters are also investigated.