This paper proposes an inertia-driven resonant piezoelectric stack pump based on a flexible support structure to solve the problem that the piezoelectric stack cannot effectively drive the diaphragm pump to transport liquid due to too small output displacement and too high resonant frequency when one end is fixed. Under the inertial force generated by the vibration of the piezoelectric stack’s mass center during its deformation, the whole piezoelectric stack will vibrate with the flexible support structure; and a large displacement and inertial force can be achieved to drive the pump at the resonant frequency. Piezoelectric pumps are designed with a diaphragm pump and a piezoelectric stack based on the flexible support structure. The piezoelectric vibrator includes a piezoelectric stack, a preloading component and a flexible support plate. A fixed support plate and three flexible support plates with different stiffnesses were fabricated and assembled with the same piezoelectric stack and diaphragm pump respectively to construct four piezoelectric pump prototypes with different resonant frequencies. The temperature rise characteristics of the piezoelectric stack were experimentally studied to determine the safe range of the driving voltage and frequency. Then the output performances of the piezoelectric pumps were tested. Under a sinusoidal driving voltage of 100 Vpp, the piezoelectric pump based on the fixed support structure cannot pump water, while the piezoelectric pumps based on the flexible support structure achieved the maximum flow rates of 89.0 ml/min, 123.4 ml/min and 197.4 ml/min at the resonant frequencies of 262 Hz, 297 Hz and 354 Hz, and the maximum backpressures of 4.4 kPa, 7.5 kPa and 11.0 kPa at 266 Hz, 309 Hz and 365 Hz.