A unique adaptive neural network control scheme is proposed for active suspension systems by employing bioinspired nonlinear dynamics, so as to address several critical engineering issues including energy efficiency, input delay, and unknown/uncertain dynamics simultaneously. A novel constructive predictor is firstly designed to solve the effect of input delay. Neural networks are then adopted to approximate the uncertain/unknown dynamics, and importantly, a unique finite-time adaptive control is established which can not only online update the input and output weights of the neural networks, but also intentionally introduce beneficial nonlinear dynamics to vibration control. The significant difference from most existing controllers lies in that, the designed controller effectively utilizes beneficial nonlinear stiffness and damping characteristics of a novel bioinspired reference model, and this leads to superior vibration suppression with significant energy-saving performance consequently. Theoretical analysis and experimental results vindicate that the proposed controller can effectively suppress vibration with much more improved control performance and fairly reduced control energy consumption. This should be for the first time to reveal both in theory and experiments that a superior suspension performance is achieved with simultaneously more than 44% control energy saving, by employing beneficial bioinspired nonlinear dynamics, compared to most traditional control methods.