In an acoustic metamaterial cavity with a coiling-up space at one end and a closed end at the other, which then forms a quarter-wave resonator, a low-frequency sound can be intensively confined within the cavity, due to a high effective refractive index and impedance. Motivated by the fact that the acoustic pressure difference between the regions at both sides of the closed end is large, this work proposes a metamaterial-based acoustic energy harvester with a piezoelectric material attached to the back plate at the closed end. To design the acoustic energy harvesting system, multi-physics simulation for electrical power output is employed using the 1/4 symmetric property to reduce computational burden, where the acoustic, structural, piezoelectric and electrical circuit analyses are fully coupled. Through the comparisons of the open-circuit voltage and output power at resonances for varying values of load resistance, it is shown that the proposed acoustic energy harvester outperforms one with no acoustic metamaterial.