The high sensitivity to impulse-type events previously uncovered for bistable oscillators has motivated recent experimental and numerical studies on the power generation performance of bistable vibration energy harvesters. To lead to an effective and efficient predictive tool and design guide, this research develops a new analytical approach to estimate the vibration response decay and power generation of a bistable energy harvester when excited by an impulse. Enabling the prediction of time-varying snap-through dynamics, this new approach greatly extends the capabilities of the current averaging method when employed with the Jacobian elliptic functions. Comparison with values determined by the direct simulation of the governing equations shows that the analytically predicted average generated power is very accurate for a wide range of impulse strengths and load resistances. The analytical approach represents a great leap forward in the everexpanding understanding of bistable vibration energy harvesters as implements to effectively capture and convert a wide range of excitation energies.